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Properties

Label 1450.2.c.e
Level $1450$
Weight $2$
Character orbit 1450.c
Analytic conductor $11.578$
Analytic rank $0$
Dimension $10$
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] = [1450,2,Mod(1101,1450)] 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("1450.1101"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1450.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,-10,0,4,0,0,-14,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.5783082931\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content 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

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)
 

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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + \beta_1 q^{3} - q^{4} - \beta_{5} q^{6} + \beta_{6} q^{7} - \beta_{3} q^{8} + (\beta_{6} + \beta_{5} - 1) q^{9} + ( - \beta_{7} + \beta_1) q^{11} - \beta_1 q^{12} + \beta_{4} q^{13}+ \cdots + (2 \beta_{7} + 2 \beta_{3} + \cdots - 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{4} + 4 q^{6} - 14 q^{9} - 4 q^{13} + 10 q^{16} - 16 q^{23} - 4 q^{24} - 16 q^{29} - 36 q^{33} + 16 q^{34} + 14 q^{36} - 24 q^{38} - 4 q^{42} + 10 q^{49} - 20 q^{51} + 4 q^{52} + 40 q^{53} - 40 q^{54}+ \cdots + 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1101.1
− 2.09023i
− 1.54715i
0.272460i
2.06121i
3.30371i
− 3.30371i
− 2.06121i
− 0.272460i
1.54715i
2.09023i
− 1.00000i − 2.09023i −1.00000 0 −2.09023 −2.45928 1.00000i −1.36905 0
1101.2 − 1.00000i − 1.54715i −1.00000 0 −1.54715 0.0591937 1.00000i 0.606340 0
1101.3 − 1.00000i 0.272460i −1.00000 0 0.272460 4.19823 1.00000i 2.92577 0
1101.4 − 1.00000i 2.06121i −1.00000 0 2.06121 1.81264 1.00000i −1.24857 0
1101.5 − 1.00000i 3.30371i −1.00000 0 3.30371 −3.61077 1.00000i −7.91448 0
1101.6 1.00000i − 3.30371i −1.00000 0 3.30371 −3.61077 − 1.00000i −7.91448 0
1101.7 1.00000i − 2.06121i −1.00000 0 2.06121 1.81264 − 1.00000i −1.24857 0
1101.8 1.00000i − 0.272460i −1.00000 0 0.272460 4.19823 − 1.00000i 2.92577 0
1101.9 1.00000i 1.54715i −1.00000 0 −1.54715 0.0591937 − 1.00000i 0.606340 0
1101.10 1.00000i 2.09023i −1.00000 0 −2.09023 −2.45928 − 1.00000i −1.36905 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1101.10
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{3}^{10} + 22T_{3}^{8} + 161T_{3}^{6} + 484T_{3}^{4} + 520T_{3}^{2} + 36 \) Copy content Toggle raw display
\( T_{7}^{5} - 20T_{7}^{3} - 6T_{7}^{2} + 68T_{7} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} + 22 T^{8} + \cdots + 36 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( (T^{5} - 20 T^{3} - 6 T^{2} + \cdots - 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + 82 T^{8} + \cdots + 270400 \) Copy content Toggle raw display
$13$ \( (T^{5} + 2 T^{4} + \cdots - 288)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + 190 T^{8} + \cdots + 51494976 \) Copy content Toggle raw display
$19$ \( T^{10} + 50 T^{8} + \cdots + 4 \) Copy content Toggle raw display
$23$ \( (T^{5} + 8 T^{4} + \cdots - 408)^{2} \) 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
$37$ \( T^{10} + 203 T^{8} + \cdots + 1089936 \) Copy content Toggle raw display
$41$ \( T^{10} + 130 T^{8} + \cdots + 44100 \) Copy content Toggle raw display
$43$ \( T^{10} + 160 T^{8} + \cdots + 16384 \) Copy content Toggle raw display
$47$ \( T^{10} + 247 T^{8} + \cdots + 171396 \) Copy content Toggle raw display
$53$ \( (T^{5} - 20 T^{4} + \cdots + 17516)^{2} \) 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
$67$ \( (T^{5} + 11 T^{4} - 10 T^{3} + \cdots + 1)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + 14 T^{4} + \cdots + 9972)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 146 T^{8} + \cdots + 15007876 \) Copy content Toggle raw display
$79$ \( T^{10} + 244 T^{8} + \cdots + 1032256 \) Copy content Toggle raw display
$83$ \( (T^{5} + 12 T^{4} + \cdots - 3164)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + 202 T^{8} + \cdots + 6140484 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 288728064 \) Copy content Toggle raw display
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