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Properties

Label 1225.4.a.bq
Level $1225$
Weight $4$
Character orbit 1225.a
Self dual yes
Analytic conductor $72.277$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1225,4,Mod(1,1225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1225, 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("1225.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1225.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: \(72.2773397570\)
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: \( x^{10} - 55x^{8} + 1007x^{6} - 6645x^{4} + 8636x^{2} - 3136 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 35)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} + 3) q^{4} + ( - \beta_{3} + \beta_{2} + 5) q^{6} + (\beta_{8} + \beta_{5} + \cdots + 2 \beta_1) q^{8} + ( - \beta_{9} - \beta_{3} + 8) q^{9} + (2 \beta_{9} - \beta_{6} - \beta_{3} + \cdots - 3) q^{11}+ \cdots + ( - 10 \beta_{9} + 62 \beta_{6} + \cdots - 678) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 30 q^{4} + 48 q^{6} + 82 q^{9} - 36 q^{11} + 22 q^{16} + 192 q^{19} + 444 q^{24} + 434 q^{26} - 130 q^{29} + 834 q^{31} - 80 q^{34} + 258 q^{36} + 868 q^{39} + 612 q^{41} + 542 q^{44} + 1274 q^{46}+ \cdots - 6930 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 55x^{8} + 1007x^{6} - 6645x^{4} + 8636x^{2} - 3136 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} - 58\nu^{6} + 1097\nu^{4} - 6968\nu^{2} + 4368 ) / 56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{9} - 163\nu^{7} + 2905\nu^{5} - 17741\nu^{3} + 11860\nu ) / 112 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{9} + 269\nu^{7} - 4743\nu^{5} + 28771\nu^{3} - 20916\nu ) / 112 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{8} - 109\nu^{6} + 1956\nu^{4} - 12109\nu^{2} + 8512 ) / 28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{9} - 381\nu^{7} + 6817\nu^{5} - 42015\nu^{3} + 30116\nu ) / 112 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{9} - 54\nu^{7} + 956\nu^{5} - 5800\nu^{3} + 3845\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3\nu^{8} - 160\nu^{6} + 2787\nu^{4} - 16606\nu^{2} + 11256 ) / 28 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{5} - \beta_{4} + 18\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} + 2\beta_{6} - 2\beta_{3} + 23\beta_{2} + 203 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 30\beta_{8} + 8\beta_{7} + 28\beta_{5} - 52\beta_{4} + 345\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -34\beta_{9} + 72\beta_{6} - 84\beta_{3} + 521\beta_{2} + 4063 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 811\beta_{8} + 296\beta_{7} + 695\beta_{5} - 1695\beta_{4} + 6952\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -875\beta_{9} + 1982\beta_{6} - 2622\beta_{3} + 11955\beta_{2} + 85243 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 20928\beta_{8} + 8336\beta_{7} + 16562\beta_{5} - 47618\beta_{4} + 146143\beta_1 \) 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
−4.87000
−4.06627
−3.63135
−0.915901
−0.850248
0.850248
0.915901
3.63135
4.06627
4.87000
−4.87000 −8.01530 15.7169 0 39.0345 0 −37.5814 37.2450 0
1.2 −4.06627 4.54560 8.53456 0 −18.4836 0 −2.17369 −6.33755 0
1.3 −3.63135 0.388534 5.18672 0 −1.41090 0 10.2160 −26.8490 0
1.4 −0.915901 −8.77417 −7.16112 0 8.03628 0 13.8861 49.9861 0
1.5 −0.850248 3.73570 −7.27708 0 −3.17627 0 12.9893 −13.0446 0
1.6 0.850248 −3.73570 −7.27708 0 −3.17627 0 −12.9893 −13.0446 0
1.7 0.915901 8.77417 −7.16112 0 8.03628 0 −13.8861 49.9861 0
1.8 3.63135 −0.388534 5.18672 0 −1.41090 0 −10.2160 −26.8490 0
1.9 4.06627 −4.54560 8.53456 0 −18.4836 0 2.17369 −6.33755 0
1.10 4.87000 8.01530 15.7169 0 39.0345 0 37.5814 37.2450 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
\(5\) \( -1 \)
\(7\) \( -1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1225.4.a.bq 10
5.b even 2 1 inner 1225.4.a.bq 10
5.c odd 4 2 245.4.b.f 10
7.b odd 2 1 1225.4.a.bp 10
7.d odd 6 2 175.4.e.g 20
35.c odd 2 1 1225.4.a.bp 10
35.f even 4 2 245.4.b.e 10
35.i odd 6 2 175.4.e.g 20
35.k even 12 4 35.4.j.a ✓ 20
35.l odd 12 4 245.4.j.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.j.a ✓ 20 35.k even 12 4
175.4.e.g 20 7.d odd 6 2
175.4.e.g 20 35.i odd 6 2
245.4.b.e 10 35.f even 4 2
245.4.b.f 10 5.c odd 4 2
245.4.j.d 20 35.l odd 12 4
1225.4.a.bp 10 7.b odd 2 1
1225.4.a.bp 10 35.c odd 2 1
1225.4.a.bq 10 1.a even 1 1 trivial
1225.4.a.bq 10 5.b 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1225))\):

\( T_{2}^{10} - 55T_{2}^{8} + 1007T_{2}^{6} - 6645T_{2}^{4} + 8636T_{2}^{2} - 3136 \) Copy content Toggle raw display
\( T_{3}^{10} - 176T_{3}^{8} + 10150T_{3}^{6} - 213472T_{3}^{4} + 1458185T_{3}^{2} - 215296 \) Copy content Toggle raw display
\( T_{19}^{5} - 96T_{19}^{4} - 4015T_{19}^{3} + 505490T_{19}^{2} - 2712444T_{19} - 282876216 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 55 T^{8} + \cdots - 3136 \) Copy content Toggle raw display
$3$ \( T^{10} - 176 T^{8} + \cdots - 215296 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( (T^{5} + 18 T^{4} + \cdots - 8279040)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 3071819776 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots - 14\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T^{5} - 96 T^{4} + \cdots - 282876216)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots - 97\!\cdots\!09 \) Copy content Toggle raw display
$29$ \( (T^{5} + 65 T^{4} + \cdots - 1210995800)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 417 T^{4} + \cdots + 115393724432)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{5} - 306 T^{4} + \cdots - 30026757530)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 26\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 37\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots - 77\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots + 1162994045920)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 1787182936154)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots - 28\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{5} + 736 T^{4} + \cdots - 445136828288)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 93\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 197075426883344)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 22\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 239387576587026)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 56\!\cdots\!24 \) Copy content Toggle raw display
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