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Properties

Label 8025.2.a.bc
Level $8025$
Weight $2$
Character orbit 8025.a
Self dual yes
Analytic conductor $64.080$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8025,2,Mod(1,8025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8025 = 3 \cdot 5^{2} \cdot 107 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8025.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: \(64.0799476221\)
Analytic rank: \(1\)
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: \( x^{10} - 2x^{9} - 13x^{8} + 26x^{7} + 51x^{6} - 101x^{5} - 65x^{4} + 126x^{3} + 5x^{2} - 10x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1605)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} - \beta_1 q^{6} + ( - \beta_{7} - \beta_{5}) q^{7} + (\beta_{5} - \beta_{4} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} - \beta_1 q^{6} + ( - \beta_{7} - \beta_{5}) q^{7} + (\beta_{5} - \beta_{4} - \beta_1) q^{8} + q^{9} + ( - \beta_{9} + \beta_{6} - \beta_{2} + \cdots - 2) q^{11}+ \cdots + ( - \beta_{9} + \beta_{6} - \beta_{2} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{2} + 10 q^{3} + 10 q^{4} - 2 q^{6} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{2} + 10 q^{3} + 10 q^{4} - 2 q^{6} + 10 q^{9} - 14 q^{11} + 10 q^{12} + 3 q^{13} - 16 q^{14} + 10 q^{16} - 8 q^{17} - 2 q^{18} - 19 q^{19} - 5 q^{22} - 4 q^{23} - 22 q^{26} + 10 q^{27} - 25 q^{29} - 2 q^{31} + 13 q^{32} - 14 q^{33} - 37 q^{34} + 10 q^{36} + 10 q^{37} + 13 q^{38} + 3 q^{39} - 31 q^{41} - 16 q^{42} - 62 q^{44} + 2 q^{46} + q^{47} + 10 q^{48} + 26 q^{49} - 8 q^{51} + 30 q^{52} - 9 q^{53} - 2 q^{54} - 63 q^{56} - 19 q^{57} - 30 q^{58} - 65 q^{59} + 12 q^{61} + 39 q^{62} - 2 q^{64} - 5 q^{66} + 10 q^{67} - 22 q^{68} - 4 q^{69} - 45 q^{71} - q^{73} + 19 q^{74} - 39 q^{76} + q^{77} - 22 q^{78} - 47 q^{79} + 10 q^{81} + 23 q^{82} - q^{83} + 12 q^{86} - 25 q^{87} + 8 q^{88} - 34 q^{89} - 26 q^{91} - 14 q^{92} - 2 q^{93} - 64 q^{94} + 13 q^{96} - 5 q^{97} + 51 q^{98} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 13x^{8} + 26x^{7} + 51x^{6} - 101x^{5} - 65x^{4} + 126x^{3} + 5x^{2} - 10x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{7} - \nu^{6} - 11\nu^{5} + 10\nu^{4} + 31\nu^{3} - 20\nu^{2} - 21\nu + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{9} - \nu^{8} - 14\nu^{7} + 14\nu^{6} + 62\nu^{5} - 60\nu^{4} - 94\nu^{3} + 83\nu^{2} + 21\nu - 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{9} - \nu^{8} - 14\nu^{7} + 14\nu^{6} + 62\nu^{5} - 60\nu^{4} - 95\nu^{3} + 83\nu^{2} + 26\nu - 6 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{9} + 2\nu^{8} + 13\nu^{7} - 25\nu^{6} - 52\nu^{5} + 91\nu^{4} + 75\nu^{3} - 104\nu^{2} - 25\nu + 6 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{9} - 2\nu^{8} - 14\nu^{7} + 27\nu^{6} + 62\nu^{5} - 111\nu^{4} - 96\nu^{3} + 145\nu^{2} + 26\nu - 7 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( 2\nu^{9} - 4\nu^{8} - 26\nu^{7} + 51\nu^{6} + 104\nu^{5} - 192\nu^{4} - 150\nu^{3} + 230\nu^{2} + 51\nu - 14 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( 5\nu^{9} - 9\nu^{8} - 66\nu^{7} + 116\nu^{6} + 269\nu^{5} - 443\nu^{4} - 385\nu^{3} + 536\nu^{2} + 108\nu - 29 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + 6\beta_{2} - \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{8} - \beta_{7} + \beta_{6} - 10\beta_{5} + 10\beta_{4} - \beta_{3} - \beta_{2} + 29\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{9} - 9\beta_{8} - 10\beta_{7} + 12\beta_{6} - 10\beta_{5} - 10\beta_{3} + 38\beta_{2} - 11\beta _1 + 86 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12\beta_{8} - 11\beta_{7} + 13\beta_{6} - 79\beta_{5} + 79\beta_{4} - 10\beta_{3} - 13\beta_{2} + 184\beta _1 - 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 79 \beta_{9} - 66 \beta_{8} - 80 \beta_{7} + 105 \beta_{6} - 77 \beta_{5} - \beta_{4} - 79 \beta_{3} + \cdots + 538 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - \beta_{9} + 106 \beta_{8} - 92 \beta_{7} + 117 \beta_{6} - 577 \beta_{5} + 580 \beta_{4} - 77 \beta_{3} + \cdots - 171 \) 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
2.59935
2.07812
1.91855
1.47926
0.173275
0.148156
−0.329799
−1.67234
−1.72711
−2.66746
−2.59935 1.00000 4.75664 0 −2.59935 4.67552 −7.16549 1.00000 0
1.2 −2.07812 1.00000 2.31860 0 −2.07812 −0.151414 −0.662092 1.00000 0
1.3 −1.91855 1.00000 1.68082 0 −1.91855 −2.92362 0.612371 1.00000 0
1.4 −1.47926 1.00000 0.188212 0 −1.47926 1.25499 2.68011 1.00000 0
1.5 −0.173275 1.00000 −1.96998 0 −0.173275 −1.72833 0.687897 1.00000 0
1.6 −0.148156 1.00000 −1.97805 0 −0.148156 0.985585 0.589370 1.00000 0
1.7 0.329799 1.00000 −1.89123 0 0.329799 0.941793 −1.28333 1.00000 0
1.8 1.67234 1.00000 0.796721 0 1.67234 −4.55690 −2.01229 1.00000 0
1.9 1.72711 1.00000 0.982925 0 1.72711 5.06713 −1.75660 1.00000 0
1.10 2.66746 1.00000 5.11534 0 2.66746 −3.56475 8.31005 1.00000 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
\(3\) \(-1\)
\(5\) \(1\)
\(107\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8025.2.a.bc 10
5.b even 2 1 1605.2.a.k ✓ 10
15.d odd 2 1 4815.2.a.p 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1605.2.a.k ✓ 10 5.b even 2 1
4815.2.a.p 10 15.d odd 2 1
8025.2.a.bc 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8025))\):

\( T_{2}^{10} + 2T_{2}^{9} - 13T_{2}^{8} - 26T_{2}^{7} + 51T_{2}^{6} + 101T_{2}^{5} - 65T_{2}^{4} - 126T_{2}^{3} + 5T_{2}^{2} + 10T_{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{10} - 48T_{7}^{8} - 22T_{7}^{7} + 698T_{7}^{6} + 494T_{7}^{5} - 2904T_{7}^{4} - 470T_{7}^{3} + 4219T_{7}^{2} - 1626T_{7} - 343 \) Copy content Toggle raw display
\( T_{11}^{10} + 14 T_{11}^{9} + 13 T_{11}^{8} - 653 T_{11}^{7} - 3089 T_{11}^{6} + 2709 T_{11}^{5} + \cdots + 53491 \) Copy content Toggle raw display
\( T_{13}^{10} - 3 T_{13}^{9} - 94 T_{13}^{8} + 204 T_{13}^{7} + 3001 T_{13}^{6} - 4210 T_{13}^{5} + \cdots - 3008 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - 48 T^{8} + \cdots - 343 \) Copy content Toggle raw display
$11$ \( T^{10} + 14 T^{9} + \cdots + 53491 \) Copy content Toggle raw display
$13$ \( T^{10} - 3 T^{9} + \cdots - 3008 \) Copy content Toggle raw display
$17$ \( T^{10} + 8 T^{9} + \cdots - 186269 \) Copy content Toggle raw display
$19$ \( T^{10} + 19 T^{9} + \cdots + 122563 \) Copy content Toggle raw display
$23$ \( T^{10} + 4 T^{9} + \cdots - 231872 \) Copy content Toggle raw display
$29$ \( T^{10} + 25 T^{9} + \cdots + 2391067 \) Copy content Toggle raw display
$31$ \( T^{10} + 2 T^{9} + \cdots + 29876672 \) Copy content Toggle raw display
$37$ \( T^{10} - 10 T^{9} + \cdots + 912064 \) Copy content Toggle raw display
$41$ \( T^{10} + 31 T^{9} + \cdots - 2556101 \) Copy content Toggle raw display
$43$ \( T^{10} - 249 T^{8} + \cdots - 3098479 \) Copy content Toggle raw display
$47$ \( T^{10} - T^{9} + \cdots - 20254208 \) Copy content Toggle raw display
$53$ \( T^{10} + 9 T^{9} + \cdots + 2074304 \) Copy content Toggle raw display
$59$ \( T^{10} + 65 T^{9} + \cdots - 2580992 \) Copy content Toggle raw display
$61$ \( T^{10} - 12 T^{9} + \cdots - 85521739 \) Copy content Toggle raw display
$67$ \( T^{10} - 10 T^{9} + \cdots - 58439 \) Copy content Toggle raw display
$71$ \( T^{10} + 45 T^{9} + \cdots + 34691264 \) Copy content Toggle raw display
$73$ \( T^{10} + T^{9} + \cdots - 1737031 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 541506409 \) Copy content Toggle raw display
$83$ \( T^{10} + T^{9} + \cdots - 356032 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 1317882551 \) Copy content Toggle raw display
$97$ \( T^{10} + 5 T^{9} + \cdots - 18130411 \) Copy content Toggle raw display
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