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Properties

Label 9025.2.a.cl
Level $9025$
Weight $2$
Character orbit 9025.a
Self dual yes
Analytic conductor $72.065$
Analytic rank $1$
Dimension $16$
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] = [9025,2,Mod(1,9025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("9025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.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: \(72.0649878242\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 22x^{14} + 190x^{12} - 820x^{10} + 1862x^{8} - 2154x^{6} + 1163x^{4} - 256x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1805)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{10} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{3} - 1) q^{6} - \beta_{15} q^{7} + (\beta_{13} + \beta_{11} + \beta_{8}) q^{8} + (\beta_{12} - \beta_{7} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{10} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{3} - 1) q^{6} - \beta_{15} q^{7} + (\beta_{13} + \beta_{11} + \beta_{8}) q^{8} + (\beta_{12} - \beta_{7} + 1) q^{9} + ( - \beta_{4} - \beta_{2} - 2) q^{11} + ( - \beta_{14} - \beta_{10} + \cdots - \beta_1) q^{12}+ \cdots + ( - 3 \beta_{12} + 5 \beta_{7} + \cdots - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9} - 22 q^{11} + 6 q^{14} + 8 q^{16} - 20 q^{21} - 14 q^{24} - 16 q^{26} + 2 q^{29} + 16 q^{31} - 8 q^{34} + 18 q^{36} - 36 q^{39} + 26 q^{41} - 64 q^{44} + 2 q^{46} - 20 q^{49} - 38 q^{51} - 12 q^{54} + 6 q^{56} + 10 q^{59} - 30 q^{61} - 16 q^{64} + 4 q^{66} + 68 q^{69} - 20 q^{71} - 40 q^{74} + 12 q^{79} - 48 q^{81} - 2 q^{84} - 20 q^{86} - 86 q^{91} - 38 q^{94} + 22 q^{96} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 22x^{14} + 190x^{12} - 820x^{10} + 1862x^{8} - 2154x^{6} + 1163x^{4} - 256x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{14} - 30\nu^{12} + 133\nu^{10} - 80\nu^{8} - 536\nu^{6} + 561\nu^{4} + 148\nu^{2} + 6 ) / 74 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{14} - 202\nu^{12} + 1342\nu^{10} - 3844\nu^{8} + 3934\nu^{6} + 1846\nu^{4} - 6327\nu^{2} + 2364 ) / 296 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{14} - 112\nu^{12} + 980\nu^{10} - 4196\nu^{8} + 8872\nu^{6} - 7940\nu^{4} + 1739\nu^{2} + 52 ) / 148 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13 \nu^{15} + 306 \nu^{13} - 2918 \nu^{11} + 14580 \nu^{9} - 40990 \nu^{7} + 63490 \nu^{5} + \cdots + 10284 \nu ) / 592 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -31\nu^{14} + 650\nu^{12} - 5262\nu^{10} + 20776\nu^{8} - 41494\nu^{6} + 39090\nu^{4} - 14393\nu^{2} + 1572 ) / 296 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -3\nu^{15} + 82\nu^{13} - 884\nu^{11} + 4782\nu^{9} - 13700\nu^{7} + 20304\nu^{5} - 14171\nu^{3} + 3210\nu ) / 148 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4\nu^{14} - 97\nu^{12} + 932\nu^{10} - 4489\nu^{8} + 11249\nu^{6} - 13678\nu^{4} + 6549\nu^{2} - 802 ) / 74 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 17 \nu^{15} + 366 \nu^{13} - 3110 \nu^{11} + 13408 \nu^{9} - 31334 \nu^{7} + 38762 \nu^{5} + \cdots + 3760 \nu ) / 296 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 31 \nu^{15} + 650 \nu^{13} - 5262 \nu^{11} + 20776 \nu^{9} - 41494 \nu^{7} + 39090 \nu^{5} + \cdots + 1572 \nu ) / 296 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -61\nu^{14} + 1322\nu^{12} - 11142\nu^{10} + 46100\nu^{8} - 96798\nu^{6} + 95610\nu^{4} - 36519\nu^{2} + 3332 ) / 296 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( \nu^{15} - 22\nu^{13} + 190\nu^{11} - 820\nu^{9} + 1862\nu^{7} - 2154\nu^{5} + 1163\nu^{3} - 248\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 73 \nu^{15} - 1650 \nu^{13} + 14678 \nu^{11} - 65228 \nu^{9} + 151302 \nu^{7} - 173570 \nu^{5} + \cdots - 12324 \nu ) / 592 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 257 \nu^{15} - 5594 \nu^{13} + 47486 \nu^{11} - 198980 \nu^{9} + 427886 \nu^{7} - 442970 \nu^{5} + \cdots - 21540 \nu ) / 592 \) 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
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{11} + \beta_{8} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{12} + 2\beta_{7} - \beta_{5} + \beta_{4} + 7\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} - \beta_{14} + 7\beta_{13} + 10\beta_{11} + 9\beta_{8} - 2\beta_{6} + 19\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -12\beta_{12} - \beta_{9} + 23\beta_{7} - 11\beta_{5} + 9\beta_{4} + \beta_{3} + 46\beta_{2} + 72 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12\beta_{15} - 14\beta_{14} + 46\beta_{13} + 80\beta_{11} + 66\beta_{8} - 22\beta_{6} + 97\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -106\beta_{12} - 14\beta_{9} + 200\beta_{7} - 88\beta_{5} + 66\beta_{4} + 14\beta_{3} + 303\beta_{2} + 391 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 106 \beta_{15} - 134 \beta_{14} + 303 \beta_{13} + 595 \beta_{11} - 6 \beta_{10} + 457 \beta_{8} + \cdots + 522 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -835\beta_{12} - 134\beta_{9} + 1568\beta_{7} - 633\beta_{5} + 457\beta_{4} + 140\beta_{3} + 2011\beta_{2} + 2214 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 835 \beta_{15} - 1109 \beta_{14} + 2011 \beta_{13} + 4280 \beta_{11} - 110 \beta_{10} + \cdots + 2933 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 6224 \beta_{12} - 1109 \beta_{9} + 11683 \beta_{7} - 4361 \beta_{5} + 3101 \beta_{4} + 1219 \beta_{3} + \cdots + 12966 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 6224 \beta_{15} - 8552 \beta_{14} + 13434 \beta_{13} + 30232 \beta_{11} - 1288 \beta_{10} + \cdots + 17075 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 45008 \beta_{12} - 8552 \beta_{9} + 84576 \beta_{7} - 29508 \beta_{5} + 20896 \beta_{4} + \cdots + 78043 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 45008 \beta_{15} - 63400 \beta_{14} + 90189 \beta_{13} + 211221 \beta_{11} - 12356 \beta_{10} + \cdots + 102328 \beta_1 \) 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
−2.61137
−2.31447
−1.93600
−1.85244
−1.24938
−0.805332
−0.578047
−0.317290
0.317290
0.578047
0.805332
1.24938
1.85244
1.93600
2.31447
2.61137
−2.61137 −0.146779 4.81924 0 0.383293 1.14057 −7.36208 −2.97846 0
1.2 −2.31447 2.90369 3.35679 0 −6.72051 −2.34671 −3.14026 5.43140 0
1.3 −1.93600 −2.41423 1.74811 0 4.67396 −1.46100 0.487655 2.82851 0
1.4 −1.85244 1.45440 1.43152 0 −2.69418 0.477604 1.05306 −0.884731 0
1.5 −1.24938 1.51315 −0.439042 0 −1.89050 0.568589 3.04730 −0.710386 0
1.6 −0.805332 −1.95838 −1.35144 0 1.57714 −1.03713 2.69902 0.835236 0
1.7 −0.578047 −0.551888 −1.66586 0 0.319017 4.66297 2.11904 −2.69542 0
1.8 −0.317290 2.04300 −1.89933 0 −0.648223 −3.69961 1.23722 1.17385 0
1.9 0.317290 −2.04300 −1.89933 0 −0.648223 3.69961 −1.23722 1.17385 0
1.10 0.578047 0.551888 −1.66586 0 0.319017 −4.66297 −2.11904 −2.69542 0
1.11 0.805332 1.95838 −1.35144 0 1.57714 1.03713 −2.69902 0.835236 0
1.12 1.24938 −1.51315 −0.439042 0 −1.89050 −0.568589 −3.04730 −0.710386 0
1.13 1.85244 −1.45440 1.43152 0 −2.69418 −0.477604 −1.05306 −0.884731 0
1.14 1.93600 2.41423 1.74811 0 4.67396 1.46100 −0.487655 2.82851 0
1.15 2.31447 −2.90369 3.35679 0 −6.72051 2.34671 3.14026 5.43140 0
1.16 2.61137 0.146779 4.81924 0 0.383293 −1.14057 7.36208 −2.97846 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(19\) \(-1\)

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 9025.2.a.cl 16
5.b even 2 1 inner 9025.2.a.cl 16
5.c odd 4 2 1805.2.b.i ✓ 16
19.b odd 2 1 9025.2.a.ck 16
95.d odd 2 1 9025.2.a.ck 16
95.g even 4 2 1805.2.b.j yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.2.b.i ✓ 16 5.c odd 4 2
1805.2.b.j yes 16 95.g even 4 2
9025.2.a.ck 16 19.b odd 2 1
9025.2.a.ck 16 95.d odd 2 1
9025.2.a.cl 16 1.a even 1 1 trivial
9025.2.a.cl 16 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9025))\):

\( T_{2}^{16} - 22T_{2}^{14} + 190T_{2}^{12} - 820T_{2}^{10} + 1862T_{2}^{8} - 2154T_{2}^{6} + 1163T_{2}^{4} - 256T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{3}^{16} - 27T_{3}^{14} + 291T_{3}^{12} - 1612T_{3}^{10} + 4892T_{3}^{8} - 7920T_{3}^{6} + 5951T_{3}^{4} - 1285T_{3}^{2} + 25 \) Copy content Toggle raw display
\( T_{7}^{16} - 46T_{7}^{14} + 709T_{7}^{12} - 4510T_{7}^{10} + 13032T_{7}^{8} - 18286T_{7}^{6} + 12345T_{7}^{4} - 3590T_{7}^{2} + 361 \) Copy content Toggle raw display
\( T_{11}^{8} + 11T_{11}^{7} + 14T_{11}^{6} - 189T_{11}^{5} - 466T_{11}^{4} + 953T_{11}^{3} + 2599T_{11}^{2} - 1702T_{11} - 3716 \) Copy content Toggle raw display
\( T_{29}^{8} - T_{29}^{7} - 116T_{29}^{6} + 381T_{29}^{5} + 2570T_{29}^{4} - 14309T_{29}^{3} + 16029T_{29}^{2} + 16054T_{29} - 27484 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 22 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{16} - 27 T^{14} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 46 T^{14} + \cdots + 361 \) Copy content Toggle raw display
$11$ \( (T^{8} + 11 T^{7} + \cdots - 3716)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 389036176 \) Copy content Toggle raw display
$17$ \( T^{16} - 135 T^{14} + \cdots + 10137856 \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} - 169 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( (T^{8} - T^{7} + \cdots - 27484)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 8 T^{7} + \cdots + 1577684)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 7255632400 \) Copy content Toggle raw display
$41$ \( (T^{8} - 13 T^{7} + \cdots + 54305)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 2834253058576 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 33594857521 \) Copy content Toggle raw display
$53$ \( T^{16} - 367 T^{14} + \cdots + 400 \) Copy content Toggle raw display
$59$ \( (T^{8} - 5 T^{7} + \cdots - 3274820)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 15 T^{7} + \cdots - 7156)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 3421867329241 \) Copy content Toggle raw display
$71$ \( (T^{8} + 10 T^{7} + \cdots + 606524)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 25761065687296 \) Copy content Toggle raw display
$79$ \( (T^{8} - 6 T^{7} + \cdots + 9098224)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 178917376 \) Copy content Toggle raw display
$89$ \( (T^{8} - 261 T^{6} + \cdots - 726211)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 14388396585616 \) Copy content Toggle raw display
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