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Properties

Label 7232.2.a.bn
Level $7232$
Weight $2$
Character orbit 7232.a
Self dual yes
Analytic conductor $57.748$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7232,2,Mod(1,7232)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7232.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7232, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7232 = 2^{6} \cdot 113 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7232.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,0,0,0,-10,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(57.7478107418\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 43 x^{16} + 760 x^{14} - 7095 x^{12} + 37240 x^{10} - 107142 x^{8} + 149152 x^{6} - 72200 x^{4} + \cdots - 800 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{10}\cdot 5 \)
Twist minimal: no (minimal twist has level 3616)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{8} - 1) q^{5} + \beta_{13} q^{7} + (\beta_{2} + 2) q^{9} + \beta_{4} q^{11} + ( - \beta_{9} + 1) q^{13} + ( - \beta_{17} + \beta_{16} + \cdots - \beta_1) q^{15} + (\beta_{9} - \beta_{7} + \beta_{6} + \cdots + 1) q^{17}+ \cdots + (2 \beta_{17} - \beta_{15} + \cdots + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 10 q^{5} + 32 q^{9} + 12 q^{13} + 16 q^{17} - 14 q^{21} + 44 q^{25} - 22 q^{29} + 24 q^{33} + 4 q^{37} + 50 q^{41} - 32 q^{45} + 58 q^{49} - 2 q^{53} + 46 q^{57} - 10 q^{61} + 40 q^{65} - 22 q^{69}+ \cdots + 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 43 x^{16} + 760 x^{14} - 7095 x^{12} + 37240 x^{10} - 107142 x^{8} + 149152 x^{6} - 72200 x^{4} + \cdots - 800 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 8\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11495 \nu^{17} - 269227 \nu^{15} + 783546 \nu^{13} + 27650591 \nu^{11} - 300446734 \nu^{9} + \cdots + 356590592 \nu ) / 18339472 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 38311 \nu^{16} + 1496973 \nu^{14} - 23477868 \nu^{12} + 188499941 \nu^{10} - 813126192 \nu^{8} + \cdots + 124916304 ) / 18339472 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 49249 \nu^{16} - 2308763 \nu^{14} + 44309516 \nu^{12} - 446030603 \nu^{10} + 2495176880 \nu^{8} + \cdots + 379270352 ) / 18339472 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 146271 \nu^{16} - 6396737 \nu^{14} + 114894104 \nu^{12} - 1086113333 \nu^{10} + 5718072756 \nu^{8} + \cdots + 357473024 ) / 18339472 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 208145 \nu^{16} - 8758887 \nu^{14} + 150919384 \nu^{12} - 1365814611 \nu^{10} + 6881284788 \nu^{8} + \cdots + 475545984 ) / 18339472 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 114485 \nu^{16} - 5367685 \nu^{14} + 102564350 \nu^{12} - 1022553055 \nu^{10} + 5628716050 \nu^{8} + \cdots + 506039752 ) / 9169736 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 345527 \nu^{16} + 15200109 \nu^{14} - 273996604 \nu^{12} + 2592878277 \nu^{10} + \cdots - 987428432 ) / 18339472 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 386003 \nu^{16} + 15259053 \nu^{14} - 244832248 \nu^{12} + 2046785537 \nu^{10} + \cdots - 745728240 ) / 18339472 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 494497 \nu^{17} - 21375475 \nu^{15} + 378911592 \nu^{13} - 3532479559 \nu^{11} + \cdots + 2205620256 \nu ) / 18339472 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 625911 \nu^{17} + 26806213 \nu^{15} - 470792596 \nu^{13} + 4349422309 \nu^{11} + \cdots - 2280333872 \nu ) / 18339472 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 689005 \nu^{17} - 29543943 \nu^{15} + 518978444 \nu^{13} - 4787275559 \nu^{11} + \cdots + 1389618672 \nu ) / 18339472 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 749749 \nu^{17} + 32121933 \nu^{15} - 564071506 \nu^{13} + 5205655571 \nu^{11} + \cdots - 1997103312 \nu ) / 18339472 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 784141 \nu^{17} - 33464697 \nu^{15} + 585367922 \nu^{13} - 5382515591 \nu^{11} + \cdots + 1873540592 \nu ) / 18339472 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 1003781 \nu^{17} - 42492811 \nu^{15} + 737070852 \nu^{13} - 6720679611 \nu^{11} + \cdots + 2705677168 \nu ) / 18339472 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} + \beta_{9} - 3\beta_{8} + \beta_{7} - 2\beta_{6} + \beta_{5} + 8\beta_{2} + 37 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{17} + 2\beta_{16} + \beta_{15} + 2\beta_{14} - 2\beta_{13} - \beta_{12} + 5\beta_{4} + 12\beta_{3} + 66\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 17 \beta_{11} - 5 \beta_{10} + 14 \beta_{9} - 56 \beta_{8} + 17 \beta_{7} - 34 \beta_{6} + 17 \beta_{5} + \cdots + 295 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 58 \beta_{17} + 44 \beta_{16} + 25 \beta_{15} + 39 \beta_{14} - 39 \beta_{13} - 20 \beta_{12} + \cdots + 569 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 223 \beta_{11} - 113 \beta_{10} + 156 \beta_{9} - 777 \beta_{8} + 218 \beta_{7} - 449 \beta_{6} + \cdots + 2488 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 823 \beta_{17} + 663 \beta_{16} + 416 \beta_{15} + 567 \beta_{14} - 572 \beta_{13} + \cdots + 5111 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2662 \beta_{11} - 1768 \beta_{10} + 1625 \beta_{9} - 9624 \beta_{8} + 2545 \beta_{7} - 5439 \beta_{6} + \cdots + 22067 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 10355 \beta_{17} + 8593 \beta_{16} + 5816 \beta_{15} + 7324 \beta_{14} - 7483 \beta_{13} + \cdots + 47673 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 30401 \beta_{11} - 23797 \beta_{10} + 16555 \beta_{9} - 112768 \beta_{8} + 28580 \beta_{7} + \cdots + 204626 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 122719 \beta_{17} + 103335 \beta_{16} + 74103 \beta_{15} + 88837 \beta_{14} - 92078 \beta_{13} + \cdots + 459742 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 339254 \beta_{11} - 296142 \beta_{10} + 167859 \beta_{9} - 1281901 \beta_{8} + 315468 \beta_{7} + \cdots + 1970544 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1406648 \beta_{17} + 1192386 \beta_{16} + 893619 \beta_{15} + 1037692 \beta_{14} + \cdots + 4560087 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 3737986 \beta_{11} - 3519792 \beta_{10} + 1707156 \beta_{9} - 14317567 \beta_{8} + 3454969 \beta_{7} + \cdots + 19567071 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 15806529 \beta_{17} + 13431189 \beta_{16} + 10409503 \beta_{15} + 11833642 \beta_{14} + \cdots + 46269930 \beta_1 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−3.29279
−2.83245
−2.69830
−2.64462
−2.40641
−1.83905
−0.552802
−0.519170
−0.334599
0.334599
0.519170
0.552802
1.83905
2.40641
2.64462
2.69830
2.83245
3.29279
0 −3.29279 0 2.49443 0 1.71345 0 7.84244 0
1.2 0 −2.83245 0 −0.321257 0 −2.16808 0 5.02276 0
1.3 0 −2.69830 0 −3.42881 0 2.84717 0 4.28081 0
1.4 0 −2.64462 0 −2.57142 0 4.31398 0 3.99399 0
1.5 0 −2.40641 0 −4.21710 0 −3.22264 0 2.79083 0
1.6 0 −1.83905 0 3.63208 0 −2.00576 0 0.382087 0
1.7 0 −0.552802 0 −2.89280 0 1.33260 0 −2.69441 0
1.8 0 −0.519170 0 1.54310 0 −4.60565 0 −2.73046 0
1.9 0 −0.334599 0 0.761772 0 4.50005 0 −2.88804 0
1.10 0 0.334599 0 0.761772 0 −4.50005 0 −2.88804 0
1.11 0 0.519170 0 1.54310 0 4.60565 0 −2.73046 0
1.12 0 0.552802 0 −2.89280 0 −1.33260 0 −2.69441 0
1.13 0 1.83905 0 3.63208 0 2.00576 0 0.382087 0
1.14 0 2.40641 0 −4.21710 0 3.22264 0 2.79083 0
1.15 0 2.64462 0 −2.57142 0 −4.31398 0 3.99399 0
1.16 0 2.69830 0 −3.42881 0 −2.84717 0 4.28081 0
1.17 0 2.83245 0 −0.321257 0 2.16808 0 5.02276 0
1.18 0 3.29279 0 2.49443 0 −1.71345 0 7.84244 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(113\) \( +1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7232.2.a.bn 18
4.b odd 2 1 inner 7232.2.a.bn 18
8.b even 2 1 3616.2.a.j ✓ 18
8.d odd 2 1 3616.2.a.j ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3616.2.a.j ✓ 18 8.b even 2 1
3616.2.a.j ✓ 18 8.d odd 2 1
7232.2.a.bn 18 1.a even 1 1 trivial
7232.2.a.bn 18 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7232))\):

\( T_{3}^{18} - 43 T_{3}^{16} + 760 T_{3}^{14} - 7095 T_{3}^{12} + 37240 T_{3}^{10} - 107142 T_{3}^{8} + \cdots - 800 \) Copy content Toggle raw display
\( T_{5}^{9} + 5T_{5}^{8} - 21T_{5}^{7} - 117T_{5}^{6} + 120T_{5}^{5} + 804T_{5}^{4} - 276T_{5}^{3} - 1740T_{5}^{2} + 640T_{5} + 368 \) Copy content Toggle raw display
\( T_{7}^{18} - 92 T_{7}^{16} + 3517 T_{7}^{14} - 72717 T_{7}^{12} + 889704 T_{7}^{10} - 6648176 T_{7}^{8} + \cdots - 66355200 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} - 43 T^{16} + \cdots - 800 \) Copy content Toggle raw display
$5$ \( (T^{9} + 5 T^{8} + \cdots + 368)^{2} \) Copy content Toggle raw display
$7$ \( T^{18} - 92 T^{16} + \cdots - 66355200 \) Copy content Toggle raw display
$11$ \( T^{18} - 140 T^{16} + \cdots - 24780800 \) Copy content Toggle raw display
$13$ \( (T^{9} - 6 T^{8} + \cdots - 6016)^{2} \) Copy content Toggle raw display
$17$ \( (T^{9} - 8 T^{8} + \cdots - 93504)^{2} \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots - 302580000 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 28752020000 \) Copy content Toggle raw display
$29$ \( (T^{9} + 11 T^{8} + \cdots + 14320)^{2} \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots - 164049920000 \) Copy content Toggle raw display
$37$ \( (T^{9} - 2 T^{8} + \cdots + 5552)^{2} \) Copy content Toggle raw display
$41$ \( (T^{9} - 25 T^{8} + \cdots + 176160)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots - 5741632800 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 22868543463200 \) Copy content Toggle raw display
$53$ \( (T^{9} + T^{8} + \cdots - 204672)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 632002759200 \) Copy content Toggle raw display
$61$ \( (T^{9} + 5 T^{8} + \cdots - 11036160)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots - 1204910784800 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 13174460467200 \) Copy content Toggle raw display
$73$ \( (T^{9} - 31 T^{8} + \cdots + 18567616)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots - 58064105319200 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 597196800 \) Copy content Toggle raw display
$89$ \( (T^{9} - 26 T^{8} + \cdots + 10969920)^{2} \) Copy content Toggle raw display
$97$ \( (T^{9} - 24 T^{8} + \cdots - 4096)^{2} \) Copy content Toggle raw display
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