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Properties

Label 2450.4.a.dc
Level $2450$
Weight $4$
Character orbit 2450.a
Self dual yes
Analytic conductor $144.555$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 214x^{8} + 15801x^{6} - 479776x^{4} + 5017216x^{2} - 1411200 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{2}\cdot 7^{2} \)
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 a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + \beta_1 q^{3} + 4 q^{4} + 2 \beta_1 q^{6} + 8 q^{8} + (\beta_{2} + 16) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + \beta_1 q^{3} + 4 q^{4} + 2 \beta_1 q^{6} + 8 q^{8} + (\beta_{2} + 16) q^{9} + ( - \beta_{7} + 5) q^{11} + 4 \beta_1 q^{12} + ( - \beta_{9} + \beta_{8} + \beta_1) q^{13} + 16 q^{16} + ( - 2 \beta_{9} - \beta_{5} + 2 \beta_{3}) q^{17} + (2 \beta_{2} + 32) q^{18} + ( - \beta_{9} - \beta_{8} + \cdots + 7 \beta_1) q^{19}+ \cdots + ( - 22 \beta_{7} - 3 \beta_{6} + \cdots - 34) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 20 q^{2} + 40 q^{4} + 80 q^{8} + 158 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 20 q^{2} + 40 q^{4} + 80 q^{8} + 158 q^{9} + 52 q^{11} + 160 q^{16} + 316 q^{18} + 104 q^{22} + 400 q^{23} + 108 q^{29} + 320 q^{32} + 632 q^{36} + 1492 q^{37} + 252 q^{39} + 904 q^{43} + 208 q^{44} + 800 q^{46} - 148 q^{51} + 968 q^{53} + 3024 q^{57} + 216 q^{58} + 640 q^{64} + 1880 q^{67} - 936 q^{71} + 1264 q^{72} + 2984 q^{74} + 504 q^{78} + 3212 q^{79} + 1010 q^{81} + 1808 q^{86} + 416 q^{88} + 1600 q^{92} + 304 q^{93} - 312 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 214x^{8} + 15801x^{6} - 479776x^{4} + 5017216x^{2} - 1411200 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 43 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} - 204\nu^{7} + 13761\nu^{5} - 354226\nu^{3} + 2765376\nu ) / 144720 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{8} + 341\nu^{6} - 17941\nu^{4} + 338344\nu^{2} - 1160208 ) / 16884 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{9} + 615\nu^{7} - 23286\nu^{5} + 51511\nu^{3} + 4085856\nu ) / 101304 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19\nu^{8} - 3474\nu^{6} + 195531\nu^{4} - 3606352\nu^{2} + 7949088 ) / 101304 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -53\nu^{8} + 8802\nu^{6} - 416577\nu^{4} + 5602448\nu^{2} - 2515464 ) / 101304 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{9} - 204\nu^{7} + 14565\nu^{5} - 462766\nu^{3} + 5820576\nu ) / 67536 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 23\nu^{9} - 4022\nu^{7} + 214663\nu^{5} - 4026028\nu^{3} + 21184608\nu ) / 337680 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 43 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{9} - 3\beta_{8} + 3\beta_{5} - 6\beta_{3} + 64\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{7} - 3\beta_{6} - 18\beta_{4} + 88\beta_{2} + 2857 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 405\beta_{9} - 321\beta_{8} + 405\beta_{5} - 990\beta_{3} + 4840\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 321\beta_{7} - 753\beta_{6} - 2610\beta_{4} + 7744\beta_{2} + 220699 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 43611\beta_{9} - 30339\beta_{8} + 43107\beta_{5} - 118542\beta_{3} + 405328\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 27819\beta_{7} - 101475\beta_{6} - 291978\beta_{4} + 700120\beta_{2} + 18694753 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4386117\beta_{9} - 2834553\beta_{8} + 4283301\beta_{5} - 12539814\beta_{3} + 35988760\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−9.74678
−7.11026
−6.79189
−4.69313
−0.537767
0.537767
4.69313
6.79189
7.11026
9.74678
2.00000 −9.74678 4.00000 0 −19.4936 0 8.00000 67.9997 0
1.2 2.00000 −7.11026 4.00000 0 −14.2205 0 8.00000 23.5558 0
1.3 2.00000 −6.79189 4.00000 0 −13.5838 0 8.00000 19.1298 0
1.4 2.00000 −4.69313 4.00000 0 −9.38627 0 8.00000 −4.97448 0
1.5 2.00000 −0.537767 4.00000 0 −1.07553 0 8.00000 −26.7108 0
1.6 2.00000 0.537767 4.00000 0 1.07553 0 8.00000 −26.7108 0
1.7 2.00000 4.69313 4.00000 0 9.38627 0 8.00000 −4.97448 0
1.8 2.00000 6.79189 4.00000 0 13.5838 0 8.00000 19.1298 0
1.9 2.00000 7.11026 4.00000 0 14.2205 0 8.00000 23.5558 0
1.10 2.00000 9.74678 4.00000 0 19.4936 0 8.00000 67.9997 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2450.4.a.dc 10
5.b even 2 1 2450.4.a.db 10
5.c odd 4 2 490.4.c.g ✓ 20
7.b odd 2 1 inner 2450.4.a.dc 10
35.c odd 2 1 2450.4.a.db 10
35.f even 4 2 490.4.c.g ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.4.c.g ✓ 20 5.c odd 4 2
490.4.c.g ✓ 20 35.f even 4 2
2450.4.a.db 10 5.b even 2 1
2450.4.a.db 10 35.c odd 2 1
2450.4.a.dc 10 1.a even 1 1 trivial
2450.4.a.dc 10 7.b odd 2 1 inner

Hecke kernels

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

\( T_{3}^{10} - 214T_{3}^{8} + 15801T_{3}^{6} - 479776T_{3}^{4} + 5017216T_{3}^{2} - 1411200 \) Copy content Toggle raw display
\( T_{11}^{5} - 26T_{11}^{4} - 4311T_{11}^{3} + 99164T_{11}^{2} - 546224T_{11} + 575456 \) Copy content Toggle raw display
\( T_{19}^{10} - 25424 T_{19}^{8} + 218838208 T_{19}^{6} - 700579963008 T_{19}^{4} + 563374160025600 T_{19}^{2} - 86\!\cdots\!00 \) Copy content Toggle raw display
\( T_{23}^{5} - 200T_{23}^{4} - 5190T_{23}^{3} + 1757712T_{23}^{2} - 22145632T_{23} - 2164142400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 214 T^{8} + \cdots - 1411200 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( (T^{5} - 26 T^{4} + \cdots + 575456)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 39\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots - 62\!\cdots\!48 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 86\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{5} - 200 T^{4} + \cdots - 2164142400)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} - 54 T^{4} + \cdots + 11338881824)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 55\!\cdots\!72 \) Copy content Toggle raw display
$37$ \( (T^{5} - 746 T^{4} + \cdots + 852756723600)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 37\!\cdots\!08 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 1418561284608)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 39\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots - 10998149389632)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 15\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 54\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots - 62326619300736)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 19001196779392)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 25\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 44582270876640)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 12\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 17\!\cdots\!68 \) Copy content Toggle raw display
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