Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1445,2,Mod(1,1445)] 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("1445.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1445 = 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1445.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,3,-3,21,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(11.5383830921\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 18 x^{10} + 55 x^{9} + 114 x^{8} - 354 x^{7} - 309 x^{6} + 936 x^{5} + 396 x^{4} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{8} q^{3} + (\beta_{2} + 2) q^{4} - q^{5} + ( - \beta_{10} + \beta_{8} + \beta_{6} + \cdots + 1) q^{6} + ( - \beta_{10} + \beta_{4} - 1) q^{7} + (\beta_{8} + \beta_{7} + \beta_{6} + \cdots + \beta_1) q^{8}+ \cdots + (\beta_{10} + \beta_{9} - \beta_{8} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} - 3 q^{3} + 21 q^{4} - 12 q^{5} + 9 q^{6} - 6 q^{7} + 12 q^{8} + 21 q^{9} - 3 q^{10} + 6 q^{11} - 6 q^{12} + 9 q^{13} + 18 q^{14} + 3 q^{15} + 39 q^{16} - 9 q^{18} + 27 q^{19} - 21 q^{20}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} - 18 x^{10} + 55 x^{9} + 114 x^{8} - 354 x^{7} - 309 x^{6} + 936 x^{5} + 396 x^{4} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8 \nu^{11} - 49 \nu^{10} - 199 \nu^{9} + 937 \nu^{8} + 1522 \nu^{7} - 5507 \nu^{6} - 4202 \nu^{5} + \cdots + 63 ) / 1332 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 7 \nu^{11} + 29 \nu^{10} + 77 \nu^{9} - 584 \nu^{8} + 139 \nu^{7} + 4000 \nu^{6} - 3344 \nu^{5} + \cdots + 486 ) / 1332 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11 \nu^{11} - 2 \nu^{10} + 232 \nu^{9} + 113 \nu^{8} - 2065 \nu^{7} - 961 \nu^{6} + 9080 \nu^{5} + \cdots - 2043 ) / 1332 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} - 57 \nu^{10} + 63 \nu^{9} + 982 \nu^{8} - 1077 \nu^{7} - 5910 \nu^{6} + 5256 \nu^{5} + \cdots - 228 ) / 444 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9 \nu^{11} + 5 \nu^{10} - 173 \nu^{9} - 42 \nu^{8} + 1111 \nu^{7} - 58 \nu^{6} - 2646 \nu^{5} + \cdots + 390 ) / 444 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 23 \nu^{11} + 127 \nu^{10} + 253 \nu^{9} - 2236 \nu^{8} - 241 \nu^{7} + 13904 \nu^{6} - 4486 \nu^{5} + \cdots - 972 ) / 1332 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 29 \nu^{11} - 86 \nu^{10} + 652 \nu^{9} + 1973 \nu^{8} - 5545 \nu^{7} - 16015 \nu^{6} + \cdots + 10053 ) / 1332 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 118 \nu^{11} + 251 \nu^{10} + 2075 \nu^{9} - 4247 \nu^{8} - 12404 \nu^{7} + 24313 \nu^{6} + \cdots - 6507 ) / 1332 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 145 \nu^{11} - 125 \nu^{10} - 2927 \nu^{9} + 2012 \nu^{8} + 21287 \nu^{7} - 10168 \nu^{6} + \cdots - 648 ) / 1332 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} - \beta_{10} + \beta_{8} + \beta_{7} - 2\beta_{5} + \beta_{4} + \beta_{3} + 8\beta_{2} + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{11} - \beta_{10} + 9\beta_{8} + 11\beta_{7} + 9\beta_{6} + 14\beta_{4} + \beta_{3} + 2\beta_{2} + 29\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 11 \beta_{11} - 10 \beta_{10} - \beta_{9} + 10 \beta_{8} + 13 \beta_{7} + \beta_{6} - 22 \beta_{5} + \cdots + 143 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 13 \beta_{11} - 11 \beta_{10} - \beta_{9} + 66 \beta_{8} + 94 \beta_{7} + 68 \beta_{6} - 8 \beta_{5} + \cdots - 7 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 95 \beta_{11} - 79 \beta_{10} - 13 \beta_{9} + 77 \beta_{8} + 129 \beta_{7} + 14 \beta_{6} + \cdots + 926 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 132 \beta_{11} - 97 \beta_{10} - 20 \beta_{9} + 454 \beta_{8} + 747 \beta_{7} + 487 \beta_{6} + \cdots - 11 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 762 \beta_{11} - 582 \beta_{10} - 128 \beta_{9} + 549 \beta_{8} + 1161 \beta_{7} + 142 \beta_{6} + \cdots + 6153 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1213 \beta_{11} - 806 \beta_{10} - 257 \beta_{9} + 3049 \beta_{8} + 5784 \beta_{7} + 3415 \beta_{6} + \cdots + 363 \) 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
−2.58439
−2.54480
−1.82030
−1.02921
−0.704111
−0.0540929
0.341036
1.66011
1.99107
2.27524
2.69854
2.77092
−2.58439 1.47257 4.67907 −1.00000 −3.80569 −4.00736 −6.92376 −0.831545 2.58439
1.2 −2.54480 −3.06453 4.47602 −1.00000 7.79863 −0.261390 −6.30099 6.39135 2.54480
1.3 −1.82030 −2.83144 1.31351 −1.00000 5.15409 −3.11716 1.24963 5.01706 1.82030
1.4 −1.02921 −1.23410 −0.940729 −1.00000 1.27015 −0.979236 3.02662 −1.47699 1.02921
1.5 −0.704111 3.11994 −1.50423 −1.00000 −2.19679 −4.06346 2.46736 6.73405 0.704111
1.6 −0.0540929 −0.605946 −1.99707 −1.00000 0.0327774 4.08179 0.216213 −2.63283 0.0540929
1.7 0.341036 1.39350 −1.88369 −1.00000 0.475233 3.70869 −1.32448 −1.05816 −0.341036
1.8 1.66011 −2.23148 0.755971 −1.00000 −3.70450 1.12916 −2.06523 1.97948 −1.66011
1.9 1.99107 −2.94118 1.96434 −1.00000 −5.85608 −3.47942 −0.0709948 5.65053 −1.99107
1.10 2.27524 1.70271 3.17670 −1.00000 3.87408 2.70724 2.67727 −0.100765 −2.27524
1.11 2.69854 2.66942 5.28213 −1.00000 7.20354 −0.656507 8.85696 4.12579 −2.69854
1.12 2.77092 −0.449467 5.67798 −1.00000 −1.24543 −1.06235 10.1914 −2.79798 −2.77092
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(17\) \( -1 \)

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 1445.2.a.r ✓ 12
5.b even 2 1 7225.2.a.bo 12
17.b even 2 1 1445.2.a.s yes 12
17.c even 4 2 1445.2.d.i 24
85.c even 2 1 7225.2.a.bn 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1445.2.a.r ✓ 12 1.a even 1 1 trivial
1445.2.a.s yes 12 17.b even 2 1
1445.2.d.i 24 17.c even 4 2
7225.2.a.bn 12 85.c even 2 1
7225.2.a.bo 12 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1445))\):

\( T_{2}^{12} - 3 T_{2}^{11} - 18 T_{2}^{10} + 55 T_{2}^{9} + 114 T_{2}^{8} - 354 T_{2}^{7} - 309 T_{2}^{6} + \cdots + 9 \) Copy content Toggle raw display
\( T_{3}^{12} + 3 T_{3}^{11} - 24 T_{3}^{10} - 71 T_{3}^{9} + 207 T_{3}^{8} + 594 T_{3}^{7} - 794 T_{3}^{6} + \cdots - 557 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 3 T^{11} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{12} + 3 T^{11} + \cdots - 557 \) Copy content Toggle raw display
$5$ \( (T + 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 6 T^{11} + \cdots + 1459 \) Copy content Toggle raw display
$11$ \( T^{12} - 6 T^{11} + \cdots + 78336 \) Copy content Toggle raw display
$13$ \( T^{12} - 9 T^{11} + \cdots - 46016 \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} - 27 T^{11} + \cdots - 8487744 \) Copy content Toggle raw display
$23$ \( T^{12} + 18 T^{11} + \cdots + 13695237 \) Copy content Toggle raw display
$29$ \( T^{12} - 198 T^{10} + \cdots - 19957023 \) Copy content Toggle raw display
$31$ \( T^{12} - 6 T^{11} + \cdots + 570816 \) Copy content Toggle raw display
$37$ \( T^{12} + 21 T^{11} + \cdots + 851392 \) Copy content Toggle raw display
$41$ \( T^{12} - 15 T^{11} + \cdots - 85203 \) Copy content Toggle raw display
$43$ \( T^{12} - 45 T^{11} + \cdots + 5779099 \) Copy content Toggle raw display
$47$ \( T^{12} - 3 T^{11} + \cdots - 32175603 \) Copy content Toggle raw display
$53$ \( T^{12} + 12 T^{11} + \cdots - 19403712 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 748413504 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 600758657 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 11025401833 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 443460096 \) Copy content Toggle raw display
$73$ \( T^{12} - 33 T^{11} + \cdots + 52193728 \) Copy content Toggle raw display
$79$ \( T^{12} - 3 T^{11} + \cdots - 43758656 \) Copy content Toggle raw display
$83$ \( T^{12} - 27 T^{11} + \cdots - 2193489 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 5665424229 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 20905823168 \) Copy content Toggle raw display
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