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Properties

Label 7248.2.a.bm
Level $7248$
Weight $2$
Character orbit 7248.a
Self dual yes
Analytic conductor $57.876$
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] = [7248,2,Mod(1,7248)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7248, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("7248.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7248 = 2^{4} \cdot 3 \cdot 151 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7248.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: \(57.8755713850\)
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: \( x^{10} - 2x^{9} - 27x^{8} + 45x^{7} + 258x^{6} - 289x^{5} - 1133x^{4} + 510x^{3} + 2070x^{2} + 341x - 500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3624)
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.

\(f(q)\) \(=\) \( q - q^{3} + \beta_1 q^{5} + ( - \beta_{2} - 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + \beta_1 q^{5} + ( - \beta_{2} - 1) q^{7} + q^{9} + ( - \beta_{8} - 1) q^{11} + (\beta_{8} - \beta_{5} + \beta_{4} + \cdots + 1) q^{13}+ \cdots + ( - \beta_{8} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{3} + 2 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{3} + 2 q^{5} - 8 q^{7} + 10 q^{9} - 7 q^{11} + 6 q^{13} - 2 q^{15} + 7 q^{17} + 8 q^{21} - 25 q^{23} + 8 q^{25} - 10 q^{27} + 12 q^{29} - 11 q^{31} + 7 q^{33} - 9 q^{35} - 3 q^{37} - 6 q^{39} + 12 q^{41} + 2 q^{45} - 31 q^{47} + 14 q^{49} - 7 q^{51} + q^{53} - 9 q^{55} - 19 q^{59} + 24 q^{61} - 8 q^{63} + 20 q^{65} + q^{67} + 25 q^{69} - 34 q^{71} - 18 q^{73} - 8 q^{75} + 27 q^{77} - 25 q^{79} + 10 q^{81} - 14 q^{83} - 3 q^{85} - 12 q^{87} + 20 q^{89} + 12 q^{91} + 11 q^{93} - 48 q^{95} - 15 q^{97} - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 27x^{8} + 45x^{7} + 258x^{6} - 289x^{5} - 1133x^{4} + 510x^{3} + 2070x^{2} + 341x - 500 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1389 \nu^{9} + 15508 \nu^{8} + 26250 \nu^{7} - 348259 \nu^{6} - 177603 \nu^{5} + 2334703 \nu^{4} + \cdots + 1342808 ) / 235018 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2633 \nu^{9} - 1651 \nu^{8} + 84784 \nu^{7} + 24672 \nu^{6} - 903653 \nu^{5} - 67662 \nu^{4} + \cdots - 374944 ) / 235018 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4033 \nu^{9} + 17956 \nu^{8} + 76979 \nu^{7} - 382350 \nu^{6} - 349183 \nu^{5} + 2276309 \nu^{4} + \cdots + 1386438 ) / 235018 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 7145 \nu^{9} + 13193 \nu^{8} + 163709 \nu^{7} - 288611 \nu^{6} - 1144545 \nu^{5} + 1768298 \nu^{4} + \cdots + 821624 ) / 235018 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 8640 \nu^{9} - 6578 \nu^{8} + 203637 \nu^{7} + 217659 \nu^{6} - 1446358 \nu^{5} - 2334780 \nu^{4} + \cdots - 3459140 ) / 235018 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8640 \nu^{9} + 6578 \nu^{8} - 203637 \nu^{7} - 217659 \nu^{6} + 1446358 \nu^{5} + 2334780 \nu^{4} + \cdots + 2049032 ) / 235018 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11279 \nu^{9} + 47420 \nu^{8} + 227115 \nu^{7} - 1039590 \nu^{6} - 1288375 \nu^{5} + \cdots + 4769416 ) / 235018 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 18007 \nu^{9} + 12050 \nu^{8} + 421267 \nu^{7} - 190178 \nu^{6} - 3008515 \nu^{5} + \cdots + 363186 ) / 235018 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + 2\beta_{8} + \beta_{6} - 2\beta_{5} + 2\beta_{3} - 3\beta_{2} + 8\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{9} + 15\beta_{7} + 12\beta_{6} - 3\beta_{5} - \beta_{4} - \beta_{3} - 2\beta _1 + 56 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 15 \beta_{9} + 34 \beta_{8} + 2 \beta_{7} + 16 \beta_{6} - 38 \beta_{5} + 5 \beta_{4} + 36 \beta_{3} + \cdots + 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 53 \beta_{9} + 3 \beta_{8} + 198 \beta_{7} + 148 \beta_{6} - 67 \beta_{5} - 11 \beta_{4} - 11 \beta_{3} + \cdots + 608 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 192 \beta_{9} + 471 \beta_{8} + 34 \beta_{7} + 207 \beta_{6} - 569 \beta_{5} + 111 \beta_{4} + \cdots + 269 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 753 \beta_{9} + 55 \beta_{8} + 2532 \beta_{7} + 1842 \beta_{6} - 1070 \beta_{5} - 64 \beta_{4} + \cdots + 7051 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2413 \beta_{9} + 6143 \beta_{8} + 380 \beta_{7} + 2513 \beta_{6} - 7812 \beta_{5} + 1821 \beta_{4} + \cdots + 1888 \) 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
−3.58272
−2.16327
−1.87603
−1.48327
−0.840171
0.415659
2.35125
2.78540
2.91109
3.48207
0 −1.00000 0 −3.58272 0 −3.12787 0 1.00000 0
1.2 0 −1.00000 0 −2.16327 0 4.13349 0 1.00000 0
1.3 0 −1.00000 0 −1.87603 0 −0.0631878 0 1.00000 0
1.4 0 −1.00000 0 −1.48327 0 −0.175714 0 1.00000 0
1.5 0 −1.00000 0 −0.840171 0 −5.06275 0 1.00000 0
1.6 0 −1.00000 0 0.415659 0 2.84379 0 1.00000 0
1.7 0 −1.00000 0 2.35125 0 −3.88084 0 1.00000 0
1.8 0 −1.00000 0 2.78540 0 −2.60389 0 1.00000 0
1.9 0 −1.00000 0 2.91109 0 0.844321 0 1.00000 0
1.10 0 −1.00000 0 3.48207 0 −0.907352 0 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
\(2\) \( +1 \)
\(3\) \( +1 \)
\(151\) \( +1 \)

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 7248.2.a.bm 10
4.b odd 2 1 3624.2.a.l ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3624.2.a.l ✓ 10 4.b odd 2 1
7248.2.a.bm 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7248))\):

\( T_{5}^{10} - 2 T_{5}^{9} - 27 T_{5}^{8} + 45 T_{5}^{7} + 258 T_{5}^{6} - 289 T_{5}^{5} - 1133 T_{5}^{4} + \cdots - 500 \) Copy content Toggle raw display
\( T_{7}^{10} + 8 T_{7}^{9} - 10 T_{7}^{8} - 203 T_{7}^{7} - 256 T_{7}^{6} + 1147 T_{7}^{5} + 2389 T_{7}^{4} + \cdots - 16 \) Copy content Toggle raw display
\( T_{11}^{10} + 7 T_{11}^{9} - 33 T_{11}^{8} - 286 T_{11}^{7} + 174 T_{11}^{6} + 3711 T_{11}^{5} + \cdots + 352 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T + 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 2 T^{9} + \cdots - 500 \) Copy content Toggle raw display
$7$ \( T^{10} + 8 T^{9} + \cdots - 16 \) Copy content Toggle raw display
$11$ \( T^{10} + 7 T^{9} + \cdots + 352 \) Copy content Toggle raw display
$13$ \( T^{10} - 6 T^{9} + \cdots + 28736 \) Copy content Toggle raw display
$17$ \( T^{10} - 7 T^{9} + \cdots + 320 \) Copy content Toggle raw display
$19$ \( T^{10} - 67 T^{8} + \cdots - 2048 \) Copy content Toggle raw display
$23$ \( T^{10} + 25 T^{9} + \cdots + 53248 \) Copy content Toggle raw display
$29$ \( T^{10} - 12 T^{9} + \cdots + 720470 \) Copy content Toggle raw display
$31$ \( T^{10} + 11 T^{9} + \cdots - 1158368 \) Copy content Toggle raw display
$37$ \( T^{10} + 3 T^{9} + \cdots - 78272 \) Copy content Toggle raw display
$41$ \( T^{10} - 12 T^{9} + \cdots + 31646488 \) Copy content Toggle raw display
$43$ \( T^{10} - 236 T^{8} + \cdots - 2858872 \) Copy content Toggle raw display
$47$ \( T^{10} + 31 T^{9} + \cdots + 1815088 \) Copy content Toggle raw display
$53$ \( T^{10} - T^{9} + \cdots + 2232704 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 103740928 \) Copy content Toggle raw display
$61$ \( T^{10} - 24 T^{9} + \cdots + 202304 \) Copy content Toggle raw display
$67$ \( T^{10} - T^{9} + \cdots - 658880 \) Copy content Toggle raw display
$71$ \( T^{10} + 34 T^{9} + \cdots - 99328 \) Copy content Toggle raw display
$73$ \( T^{10} + 18 T^{9} + \cdots + 2918656 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 6951633728 \) Copy content Toggle raw display
$83$ \( T^{10} + 14 T^{9} + \cdots - 1615360 \) Copy content Toggle raw display
$89$ \( T^{10} - 20 T^{9} + \cdots + 37271936 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 6649111736 \) Copy content Toggle raw display
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