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Properties

Label 1450.2.j.g
Level $1450$
Weight $2$
Character orbit 1450.j
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(157,1450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1450, base_ring=CyclotomicField(4))
 
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("1450.157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1450.j (of order \(4\), degree \(2\), 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: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 18x^{8} + 99x^{6} + 186x^{4} + 105x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} - \beta_{2} q^{3} - q^{4} - \beta_1 q^{6} + (\beta_{9} + \beta_{5} + \beta_{4} + \cdots - 1) q^{7}+ \cdots + ( - \beta_{3} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} - \beta_{2} q^{3} - q^{4} - \beta_1 q^{6} + (\beta_{9} + \beta_{5} + \beta_{4} + \cdots - 1) q^{7}+ \cdots + ( - 2 \beta_{9} + \beta_{8} - \beta_{5} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{3} - 10 q^{4} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 4 q^{3} - 10 q^{4} + 2 q^{7} + 6 q^{9} - 4 q^{11} - 4 q^{12} - 4 q^{13} + 2 q^{14} + 10 q^{16} - 4 q^{19} + 20 q^{21} - 4 q^{22} - 6 q^{23} - 4 q^{26} + 16 q^{27} - 2 q^{28} + 8 q^{29} - 4 q^{31} - 36 q^{33} + 8 q^{34} - 6 q^{36} + 20 q^{37} + 4 q^{38} + 12 q^{39} - 6 q^{41} + 20 q^{42} - 8 q^{43} + 4 q^{44} + 6 q^{46} + 4 q^{47} + 4 q^{48} + 4 q^{52} - 2 q^{56} + 4 q^{57} + 10 q^{58} - 6 q^{61} - 4 q^{62} + 26 q^{63} - 10 q^{64} - 36 q^{66} + 6 q^{67} - 16 q^{69} + 4 q^{76} + 12 q^{78} + 8 q^{79} + 18 q^{81} + 6 q^{82} + 14 q^{83} - 20 q^{84} + 4 q^{88} + 34 q^{89} + 6 q^{92} - 48 q^{93} - 24 q^{97} + 34 q^{98} - 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 18x^{8} + 99x^{6} + 186x^{4} + 105x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{8} - 33\nu^{6} - 155\nu^{4} - 211\nu^{2} - 56 ) / 26 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{8} + 33\nu^{6} + 155\nu^{4} + 237\nu^{2} + 134 ) / 26 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{9} + \nu^{8} + 56\nu^{7} + 23\nu^{6} + 330\nu^{5} + 175\nu^{4} + 687\nu^{3} + 450\nu^{2} + 318\nu + 184 ) / 52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -7\nu^{9} - 122\nu^{7} - 627\nu^{5} - 992\nu^{3} - 313\nu ) / 52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5 \nu^{9} - 11 \nu^{8} - 89 \nu^{7} - 188 \nu^{6} - 472 \nu^{5} - 924 \nu^{4} - 755 \nu^{3} + \cdots - 308 ) / 52 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9 \nu^{9} + 12 \nu^{8} + 155 \nu^{7} + 211 \nu^{6} + 769 \nu^{5} + 1099 \nu^{4} + 1060 \nu^{3} + \cdots + 440 ) / 52 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 17\nu^{9} + 300\nu^{7} + 1571\nu^{5} + 2554\nu^{3} + 879\nu ) / 52 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9 \nu^{9} - 12 \nu^{8} + 155 \nu^{7} - 211 \nu^{6} + 769 \nu^{5} - 1099 \nu^{4} + 1060 \nu^{3} + \cdots - 440 ) / 52 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + \beta_{8} + \beta_{6} - \beta_{4} - 8\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{7} + 2\beta_{6} + 2\beta_{5} + 2\beta_{4} - 8\beta_{3} - 13\beta_{2} + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{9} - 13\beta_{8} - 2\beta_{7} - 11\beta_{6} - 10\beta_{5} + 11\beta_{4} + 67\beta _1 - 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -15\beta_{9} - 11\beta_{7} - 26\beta_{6} - 26\beta_{5} - 26\beta_{4} + 67\beta_{3} + 144\beta_{2} - 131 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -78\beta_{9} + 144\beta_{8} + 26\beta_{7} + 104\beta_{6} + 164\beta_{5} - 104\beta_{4} - 591\beta _1 + 104 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 170\beta_{9} + 104\beta_{7} + 274\beta_{6} + 274\beta_{5} + 274\beta_{4} - 591\beta_{3} - 1487\beta_{2} + 1055 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 695\beta_{9} - 1487\beta_{8} - 274\beta_{7} - 969\beta_{6} - 1970\beta_{5} + 969\beta_{4} + 5388\beta _1 - 969 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(901\) \(1277\)
\(\chi(n)\) \(-\beta_{5}\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
− 2.32530i
− 0.752486i
0.496555i
1.48865i
3.09258i
2.32530i
0.752486i
− 0.496555i
− 1.48865i
− 3.09258i
− 1.00000i −2.32530 −1.00000 0 2.32530i −2.17677 − 2.17677i 1.00000i 2.40704 0
157.2 − 1.00000i −0.752486 −1.00000 0 0.752486i 2.38506 + 2.38506i 1.00000i −2.43376 0
157.3 − 1.00000i 0.496555 −1.00000 0 − 0.496555i −0.303453 − 0.303453i 1.00000i −2.75343 0
157.4 − 1.00000i 1.48865 −1.00000 0 − 1.48865i −2.18018 − 2.18018i 1.00000i −0.783913 0
157.5 − 1.00000i 3.09258 −1.00000 0 − 3.09258i 3.27535 + 3.27535i 1.00000i 6.56407 0
1293.1 1.00000i −2.32530 −1.00000 0 − 2.32530i −2.17677 + 2.17677i − 1.00000i 2.40704 0
1293.2 1.00000i −0.752486 −1.00000 0 − 0.752486i 2.38506 − 2.38506i − 1.00000i −2.43376 0
1293.3 1.00000i 0.496555 −1.00000 0 0.496555i −0.303453 + 0.303453i − 1.00000i −2.75343 0
1293.4 1.00000i 1.48865 −1.00000 0 1.48865i −2.18018 + 2.18018i − 1.00000i −0.783913 0
1293.5 1.00000i 3.09258 −1.00000 0 3.09258i 3.27535 − 3.27535i − 1.00000i 6.56407 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
145.e even 4 1 inner

Twists

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

Hecke kernels

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

\( T_{3}^{5} - 2T_{3}^{4} - 7T_{3}^{3} + 10T_{3}^{2} + 5T_{3} - 4 \) Copy content Toggle raw display
\( T_{11}^{10} + 4 T_{11}^{9} + 8 T_{11}^{8} - 16 T_{11}^{7} + 156 T_{11}^{6} + 448 T_{11}^{5} + 672 T_{11}^{4} + \cdots + 512 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$3$ \( (T^{5} - 2 T^{4} - 7 T^{3} + \cdots - 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - 2 T^{9} + \cdots + 4050 \) Copy content Toggle raw display
$11$ \( T^{10} + 4 T^{9} + \cdots + 512 \) Copy content Toggle raw display
$13$ \( T^{10} + 4 T^{9} + \cdots + 392 \) Copy content Toggle raw display
$17$ \( T^{10} + 82 T^{8} + \cdots + 36 \) Copy content Toggle raw display
$19$ \( T^{10} + 4 T^{9} + \cdots + 128 \) Copy content Toggle raw display
$23$ \( T^{10} + 6 T^{9} + \cdots + 1250 \) Copy content Toggle raw display
$29$ \( T^{10} - 8 T^{9} + \cdots + 20511149 \) Copy content Toggle raw display
$31$ \( T^{10} + 4 T^{9} + \cdots + 127008 \) Copy content Toggle raw display
$37$ \( (T^{5} - 10 T^{4} + \cdots + 24)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + 6 T^{9} + \cdots + 35212832 \) Copy content Toggle raw display
$43$ \( (T^{5} + 4 T^{4} + \cdots + 40750)^{2} \) Copy content Toggle raw display
$47$ \( (T^{5} - 2 T^{4} + \cdots - 1544)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} - 144 T^{7} + \cdots + 12500000 \) Copy content Toggle raw display
$59$ \( T^{10} + 182 T^{8} + \cdots + 430336 \) Copy content Toggle raw display
$61$ \( T^{10} + 6 T^{9} + \cdots + 16971138 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 2259533088 \) Copy content Toggle raw display
$71$ \( T^{10} + 392 T^{8} + \cdots + 36000000 \) Copy content Toggle raw display
$73$ \( T^{10} + 242 T^{8} + \cdots + 3118756 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 922608968 \) Copy content Toggle raw display
$83$ \( T^{10} - 14 T^{9} + \cdots + 17287200 \) Copy content Toggle raw display
$89$ \( T^{10} - 34 T^{9} + \cdots + 25088 \) Copy content Toggle raw display
$97$ \( (T^{5} + 12 T^{4} + \cdots + 44794)^{2} \) Copy content Toggle raw display
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