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Properties

Label 1450.2.d.j
Level $1450$
Weight $2$
Character orbit 1450.d
Analytic conductor $11.578$
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] = [1450,2,Mod(1449,1450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1450.1449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1450.d (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(11.5783082931\)
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: \( x^{10} + 22x^{8} + 161x^{6} + 484x^{4} + 520x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + q^{2} - \beta_{5} q^{3} + q^{4} - \beta_{5} q^{6} + \beta_{2} q^{7} + q^{8} + ( - \beta_{6} - \beta_{5} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta_{5} q^{3} + q^{4} - \beta_{5} q^{6} + \beta_{2} q^{7} + q^{8} + ( - \beta_{6} - \beta_{5} + 1) q^{9} + ( - \beta_{7} + \beta_1) q^{11} - \beta_{5} q^{12} + \beta_{7} q^{13} + \beta_{2} q^{14} + q^{16} + ( - \beta_{9} - \beta_{8} + \beta_{5} + \cdots - 2) q^{17}+ \cdots + ( - 2 \beta_{7} - 2 \beta_{3} + \cdots + 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} + 4 q^{3} + 10 q^{4} + 4 q^{6} + 10 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{2} + 4 q^{3} + 10 q^{4} + 4 q^{6} + 10 q^{8} + 14 q^{9} + 4 q^{12} + 10 q^{16} - 16 q^{17} + 14 q^{18} + 4 q^{24} + 40 q^{27} + 16 q^{29} + 10 q^{32} - 16 q^{34} + 14 q^{36} - 30 q^{37} - 10 q^{47} + 4 q^{48} - 10 q^{49} - 20 q^{51} + 40 q^{54} + 16 q^{58} - 18 q^{59} + 10 q^{64} - 16 q^{68} - 28 q^{71} + 14 q^{72} - 8 q^{73} - 30 q^{74} + 18 q^{81} + 60 q^{87} + 4 q^{91} - 10 q^{94} + 4 q^{96} - 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 22x^{8} + 161x^{6} + 484x^{4} + 520x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 17\nu^{5} + 76\nu^{3} + 98\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{9} - 18\nu^{7} - 90\nu^{5} - 141\nu^{3} - 32\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{8} - 18\nu^{6} - 89\nu^{4} - 128\nu^{2} - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{8} - 71\nu^{6} - 343\nu^{4} - 488\nu^{2} - 36 ) / 6 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4\nu^{8} + 71\nu^{6} + 343\nu^{4} + 494\nu^{2} + 60 ) / 6 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{9} + 35\nu^{7} + 163\nu^{5} + 209\nu^{3} - 13\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{9} - 11\nu^{8} - 17\nu^{7} - 193\nu^{6} - 76\nu^{5} - 908\nu^{4} - 110\nu^{3} - 1228\nu^{2} - 60\nu - 72 ) / 12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{9} - 11\nu^{8} + 17\nu^{7} - 193\nu^{6} + 76\nu^{5} - 908\nu^{4} + 110\nu^{3} - 1228\nu^{2} + 60\nu - 72 ) / 12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 4\beta_{3} + 2\beta_{2} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} - 10\beta_{6} - 15\beta_{5} + 3\beta_{4} + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{9} - 2\beta_{8} - 15\beta_{7} - 58\beta_{3} - 28\beta_{2} + 63\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -13\beta_{9} - 13\beta_{8} + 106\beta_{6} + 177\beta_{5} - 47\beta_{4} - 248 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -34\beta_{9} + 34\beta_{8} + 179\beta_{7} + 682\beta_{3} + 330\beta_{2} - 637\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 145\beta_{9} + 145\beta_{8} - 1146\beta_{6} - 1979\beta_{5} + 577\beta_{4} + 2480 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 432\beta_{9} - 432\beta_{8} - 2013\beta_{7} - 7626\beta_{3} - 3702\beta_{2} + 6751\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1450\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1277\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1449.1
− 2.09023i
2.09023i
1.54715i
− 1.54715i
− 0.272460i
0.272460i
− 2.06121i
2.06121i
3.30371i
− 3.30371i
1.00000 −2.09023 1.00000 0 −2.09023 − 2.45928i 1.00000 1.36905 0
1449.2 1.00000 −2.09023 1.00000 0 −2.09023 2.45928i 1.00000 1.36905 0
1449.3 1.00000 −1.54715 1.00000 0 −1.54715 − 0.0591937i 1.00000 −0.606340 0
1449.4 1.00000 −1.54715 1.00000 0 −1.54715 0.0591937i 1.00000 −0.606340 0
1449.5 1.00000 0.272460 1.00000 0 0.272460 − 4.19823i 1.00000 −2.92577 0
1449.6 1.00000 0.272460 1.00000 0 0.272460 4.19823i 1.00000 −2.92577 0
1449.7 1.00000 2.06121 1.00000 0 2.06121 − 1.81264i 1.00000 1.24857 0
1449.8 1.00000 2.06121 1.00000 0 2.06121 1.81264i 1.00000 1.24857 0
1449.9 1.00000 3.30371 1.00000 0 3.30371 − 3.61077i 1.00000 7.91448 0
1449.10 1.00000 3.30371 1.00000 0 3.30371 3.61077i 1.00000 7.91448 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1449.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
145.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1450.2.d.j 10
5.b even 2 1 1450.2.d.i 10
5.c odd 4 1 1450.2.c.e ✓ 10
5.c odd 4 1 1450.2.c.f yes 10
29.b even 2 1 1450.2.d.i 10
145.d even 2 1 inner 1450.2.d.j 10
145.h odd 4 1 1450.2.c.e ✓ 10
145.h odd 4 1 1450.2.c.f yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1450.2.c.e ✓ 10 5.c odd 4 1
1450.2.c.e ✓ 10 145.h odd 4 1
1450.2.c.f yes 10 5.c odd 4 1
1450.2.c.f yes 10 145.h odd 4 1
1450.2.d.i 10 5.b even 2 1
1450.2.d.i 10 29.b even 2 1
1450.2.d.j 10 1.a even 1 1 trivial
1450.2.d.j 10 145.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1450, [\chi])\):

\( T_{3}^{5} - 2T_{3}^{4} - 9T_{3}^{3} + 10T_{3}^{2} + 20T_{3} - 6 \) Copy content Toggle raw display
\( T_{17}^{5} + 8T_{17}^{4} - 63T_{17}^{3} - 514T_{17}^{2} + 860T_{17} + 7176 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{10} \) Copy content Toggle raw display
$3$ \( (T^{5} - 2 T^{4} - 9 T^{3} + \cdots - 6)^{2} \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 40 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{10} + 82 T^{8} + \cdots + 270400 \) Copy content Toggle raw display
$13$ \( T^{10} + 68 T^{8} + \cdots + 82944 \) Copy content Toggle raw display
$17$ \( (T^{5} + 8 T^{4} + \cdots + 7176)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + 50 T^{8} + \cdots + 4 \) Copy content Toggle raw display
$23$ \( T^{10} + 172 T^{8} + \cdots + 166464 \) Copy content Toggle raw display
$29$ \( T^{10} - 16 T^{9} + \cdots + 20511149 \) Copy content Toggle raw display
$31$ \( T^{10} + 123 T^{8} + \cdots + 19600 \) Copy content Toggle raw display
$37$ \( (T^{5} + 15 T^{4} + \cdots - 1044)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + 130 T^{8} + \cdots + 44100 \) Copy content Toggle raw display
$43$ \( (T^{5} - 80 T^{3} + \cdots - 128)^{2} \) Copy content Toggle raw display
$47$ \( (T^{5} + 5 T^{4} + \cdots + 414)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 306810256 \) Copy content Toggle raw display
$59$ \( (T^{5} + 9 T^{4} + \cdots - 24840)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + 167 T^{8} + \cdots + 739600 \) Copy content Toggle raw display
$67$ \( T^{10} + 141 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( (T^{5} + 14 T^{4} + \cdots + 9972)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 4 T^{4} + \cdots + 3874)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + 244 T^{8} + \cdots + 1032256 \) Copy content Toggle raw display
$83$ \( T^{10} + 442 T^{8} + \cdots + 10010896 \) Copy content Toggle raw display
$89$ \( T^{10} + 202 T^{8} + \cdots + 6140484 \) Copy content Toggle raw display
$97$ \( (T^{5} - 258 T^{3} + \cdots - 16992)^{2} \) Copy content Toggle raw display
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