Properties

Label 2450.4.a.bt
Level $2450$
Weight $4$
Character orbit 2450.a
Self dual yes
Analytic conductor $144.555$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2450,4,Mod(1,2450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2450.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2450.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(144.554679514\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{177}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 490)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{177})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + ( - \beta - 2) q^{3} + 4 q^{4} + ( - 2 \beta - 4) q^{6} + 8 q^{8} + (5 \beta + 21) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + ( - \beta - 2) q^{3} + 4 q^{4} + ( - 2 \beta - 4) q^{6} + 8 q^{8} + (5 \beta + 21) q^{9} - 5 \beta q^{11} + ( - 4 \beta - 8) q^{12} + ( - 3 \beta - 46) q^{13} + 16 q^{16} + (\beta + 12) q^{17} + (10 \beta + 42) q^{18} + ( - 4 \beta - 58) q^{19} - 10 \beta q^{22} + ( - 20 \beta - 52) q^{23} + ( - 8 \beta - 16) q^{24} + ( - 6 \beta - 92) q^{26} + ( - 9 \beta - 208) q^{27} + (25 \beta + 94) q^{29} + ( - 2 \beta + 36) q^{31} + 32 q^{32} + (15 \beta + 220) q^{33} + (2 \beta + 24) q^{34} + (20 \beta + 84) q^{36} + ( - 30 \beta + 82) q^{37} + ( - 8 \beta - 116) q^{38} + (55 \beta + 224) q^{39} + ( - 38 \beta + 204) q^{41} + (10 \beta + 36) q^{43} - 20 \beta q^{44} + ( - 40 \beta - 104) q^{46} + ( - 83 \beta - 16) q^{47} + ( - 16 \beta - 32) q^{48} + ( - 15 \beta - 68) q^{51} + ( - 12 \beta - 184) q^{52} + (20 \beta + 502) q^{53} + ( - 18 \beta - 416) q^{54} + (70 \beta + 292) q^{57} + (50 \beta + 188) q^{58} + ( - 36 \beta + 398) q^{59} + ( - 66 \beta - 362) q^{61} + ( - 4 \beta + 72) q^{62} + 64 q^{64} + (30 \beta + 440) q^{66} + ( - 10 \beta - 420) q^{67} + (4 \beta + 48) q^{68} + (112 \beta + 984) q^{69} + ( - 100 \beta + 236) q^{71} + (40 \beta + 168) q^{72} + (106 \beta - 148) q^{73} + ( - 60 \beta + 164) q^{74} + ( - 16 \beta - 232) q^{76} + (110 \beta + 448) q^{78} + ( - 55 \beta + 240) q^{79} + (100 \beta + 245) q^{81} + ( - 76 \beta + 408) q^{82} + (6 \beta - 1058) q^{83} + (20 \beta + 72) q^{86} + ( - 169 \beta - 1288) q^{87} - 40 \beta q^{88} + ( - 160 \beta + 360) q^{89} + ( - 80 \beta - 208) q^{92} + ( - 30 \beta + 16) q^{93} + ( - 166 \beta - 32) q^{94} + ( - 32 \beta - 64) q^{96} + (85 \beta - 380) q^{97} + ( - 130 \beta - 1100) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - 5 q^{3} + 8 q^{4} - 10 q^{6} + 16 q^{8} + 47 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} - 5 q^{3} + 8 q^{4} - 10 q^{6} + 16 q^{8} + 47 q^{9} - 5 q^{11} - 20 q^{12} - 95 q^{13} + 32 q^{16} + 25 q^{17} + 94 q^{18} - 120 q^{19} - 10 q^{22} - 124 q^{23} - 40 q^{24} - 190 q^{26} - 425 q^{27} + 213 q^{29} + 70 q^{31} + 64 q^{32} + 455 q^{33} + 50 q^{34} + 188 q^{36} + 134 q^{37} - 240 q^{38} + 503 q^{39} + 370 q^{41} + 82 q^{43} - 20 q^{44} - 248 q^{46} - 115 q^{47} - 80 q^{48} - 151 q^{51} - 380 q^{52} + 1024 q^{53} - 850 q^{54} + 654 q^{57} + 426 q^{58} + 760 q^{59} - 790 q^{61} + 140 q^{62} + 128 q^{64} + 910 q^{66} - 850 q^{67} + 100 q^{68} + 2080 q^{69} + 372 q^{71} + 376 q^{72} - 190 q^{73} + 268 q^{74} - 480 q^{76} + 1006 q^{78} + 425 q^{79} + 590 q^{81} + 740 q^{82} - 2110 q^{83} + 164 q^{86} - 2745 q^{87} - 40 q^{88} + 560 q^{89} - 496 q^{92} + 2 q^{93} - 230 q^{94} - 160 q^{96} - 675 q^{97} - 2330 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
7.15207
−6.15207
2.00000 −9.15207 4.00000 0 −18.3041 0 8.00000 56.7603 0
1.2 2.00000 4.15207 4.00000 0 8.30413 0 8.00000 −9.76034 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2450.4.a.bt 2
5.b even 2 1 490.4.a.u yes 2
7.b odd 2 1 2450.4.a.ca 2
35.c odd 2 1 490.4.a.p 2
35.i odd 6 2 490.4.e.x 4
35.j even 6 2 490.4.e.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.4.a.p 2 35.c odd 2 1
490.4.a.u yes 2 5.b even 2 1
490.4.e.t 4 35.j even 6 2
490.4.e.x 4 35.i odd 6 2
2450.4.a.bt 2 1.a even 1 1 trivial
2450.4.a.ca 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2450))\):

\( T_{3}^{2} + 5T_{3} - 38 \) Copy content Toggle raw display
\( T_{11}^{2} + 5T_{11} - 1100 \) Copy content Toggle raw display
\( T_{19}^{2} + 120T_{19} + 2892 \) Copy content Toggle raw display
\( T_{23}^{2} + 124T_{23} - 13856 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 5T - 38 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 5T - 1100 \) Copy content Toggle raw display
$13$ \( T^{2} + 95T + 1858 \) Copy content Toggle raw display
$17$ \( T^{2} - 25T + 112 \) Copy content Toggle raw display
$19$ \( T^{2} + 120T + 2892 \) Copy content Toggle raw display
$23$ \( T^{2} + 124T - 13856 \) Copy content Toggle raw display
$29$ \( T^{2} - 213T - 16314 \) Copy content Toggle raw display
$31$ \( T^{2} - 70T + 1048 \) Copy content Toggle raw display
$37$ \( T^{2} - 134T - 35336 \) Copy content Toggle raw display
$41$ \( T^{2} - 370T - 29672 \) Copy content Toggle raw display
$43$ \( T^{2} - 82T - 2744 \) Copy content Toggle raw display
$47$ \( T^{2} + 115T - 301532 \) Copy content Toggle raw display
$53$ \( T^{2} - 1024 T + 244444 \) Copy content Toggle raw display
$59$ \( T^{2} - 760T + 87052 \) Copy content Toggle raw display
$61$ \( T^{2} + 790T - 36728 \) Copy content Toggle raw display
$67$ \( T^{2} + 850T + 176200 \) Copy content Toggle raw display
$71$ \( T^{2} - 372T - 407904 \) Copy content Toggle raw display
$73$ \( T^{2} + 190T - 488168 \) Copy content Toggle raw display
$79$ \( T^{2} - 425T - 88700 \) Copy content Toggle raw display
$83$ \( T^{2} + 2110 T + 1111432 \) Copy content Toggle raw display
$89$ \( T^{2} - 560 T - 1054400 \) Copy content Toggle raw display
$97$ \( T^{2} + 675T - 205800 \) Copy content Toggle raw display
show more
show less