↑

Properties

Label 2252.3.d.a
Level $2252$
Weight $3$
Character orbit 2252.d
Self dual yes
Analytic conductor $61.363$
Analytic rank $0$
Dimension $18$
CM discriminant -563
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2252,3,Mod(1125,2252)] 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("2252.1125"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2252, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 2252 = 2^{2} \cdot 563 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2252.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(61.3625555339\)
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: \( x^{18} - 54 x^{16} + 1215 x^{14} - 14742 x^{12} + 104247 x^{10} - 103 x^{9} - 433026 x^{8} + \cdots - 48440 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{13} q^{7} + ( - \beta_{7} + \beta_{6} + 9) q^{9} + (\beta_{11} + 2 \beta_{10} + 2 \beta_{4}) q^{11} + (2 \beta_{11} - \beta_{8} + \beta_{4}) q^{13} + ( - 3 \beta_{12} - \beta_{7} - \beta_{6}) q^{17}+ \cdots + (\beta_{17} - 3 \beta_{16} + \cdots + 11 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 162 q^{9} + 450 q^{25} + 882 q^{49} + 1458 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 54 x^{16} + 1215 x^{14} - 14742 x^{12} + 104247 x^{10} - 103 x^{9} - 433026 x^{8} + \cdots - 48440 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{9} + 27\nu^{7} - 243\nu^{5} + 810\nu^{3} - 729\nu + 57 ) / 11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11 \nu^{15} - 495 \nu^{13} - 210 \nu^{12} + 8910 \nu^{11} + 7560 \nu^{10} - 81675 \nu^{9} + \cdots - 61182 ) / 49907 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2 \nu^{16} - 96 \nu^{14} + 1872 \nu^{12} - 527 \nu^{11} - 19008 \nu^{10} + 17391 \nu^{9} + \cdots - 978 ) / 49907 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 92 \nu^{14} - 173 \nu^{13} - 3864 \nu^{12} + 6747 \nu^{11} + 63756 \nu^{10} - 101205 \nu^{9} + \cdots - 75744 ) / 49907 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 92 \nu^{14} - 173 \nu^{13} - 3864 \nu^{12} + 6747 \nu^{11} + 63756 \nu^{10} - 101205 \nu^{9} + \cdots - 974070 ) / 49907 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 49 \nu^{15} - 2205 \nu^{13} - 523 \nu^{12} + 39690 \nu^{11} + 18828 \nu^{10} - 363825 \nu^{9} + \cdots - 1497528 ) / 49907 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 129 \nu^{14} + 4 \nu^{13} - 5418 \nu^{12} - 156 \nu^{11} + 89397 \nu^{10} + 2340 \nu^{9} + \cdots - 564246 ) / 49907 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 157 \nu^{15} + 7065 \nu^{13} + 935 \nu^{12} - 127170 \nu^{11} - 33660 \nu^{10} + 1165725 \nu^{9} + \cdots + 1608228 ) / 49907 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 189 \nu^{15} - 8505 \nu^{13} - 721 \nu^{12} + 153090 \nu^{11} + 25956 \nu^{10} - 1403325 \nu^{9} + \cdots - 1541214 ) / 49907 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 357 \nu^{14} + 622 \nu^{13} + 14994 \nu^{12} - 24258 \nu^{11} - 247401 \nu^{10} + 363870 \nu^{9} + \cdots + 1643184 ) / 49907 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 12 \nu^{17} + 612 \nu^{15} - 12852 \nu^{13} + 143208 \nu^{11} + 412 \nu^{10} - 908820 \nu^{9} + \cdots - 935226 ) / 49907 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 23 \nu^{17} - 1173 \nu^{15} + 24633 \nu^{13} - 274482 \nu^{11} - 2302 \nu^{10} + 1741905 \nu^{9} + \cdots + 1118772 ) / 49907 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 45 \nu^{16} + 2160 \nu^{14} - 42120 \nu^{12} + 515 \nu^{11} + 427680 \nu^{10} - 16995 \nu^{9} + \cdots - 862710 ) / 49907 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 3 \nu^{17} + 153 \nu^{15} - 3213 \nu^{13} + 35802 \nu^{11} + 103 \nu^{10} - 227205 \nu^{9} + \cdots - 50058 ) / 4537 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 24 \nu^{16} + 1152 \nu^{14} - 22464 \nu^{12} + 391 \nu^{11} + 228096 \nu^{10} - 12903 \nu^{9} + \cdots - 381936 ) / 3839 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{11} + 3\beta_{10} + \beta_{8} + 4\beta_{4} + 9\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} + \beta_{6} + 12\beta_{2} + 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5 \beta_{12} + 30 \beta_{11} + 45 \beta_{10} + \beta_{9} + 15 \beta_{8} + 7 \beta_{7} + 11 \beta_{6} + \cdots + 90 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{11} + 2\beta_{10} + 3\beta_{8} - 18\beta_{7} + 18\beta_{6} - 2\beta_{4} + 135\beta_{2} + 540 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 5 \beta_{17} - 34 \beta_{15} + 105 \beta_{12} + 378 \beta_{11} + 567 \beta_{10} + 21 \beta_{9} + \cdots + 945 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4 \beta_{16} - 11 \beta_{13} + 24 \beta_{11} + 48 \beta_{10} + 72 \beta_{8} - 252 \beta_{7} + \cdots + 5670 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 135 \beta_{17} - 918 \beta_{15} + 1620 \beta_{12} + 4536 \beta_{11} + 6804 \beta_{10} + 324 \beta_{9} + \cdots + 228 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 97 \beta_{16} - 33 \beta_{14} - 330 \beta_{13} + 405 \beta_{11} + 810 \beta_{10} + 1215 \beta_{8} + \cdots + 61236 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2518 \beta_{17} - 17140 \beta_{15} + 22275 \beta_{12} + 53460 \beta_{11} + 80190 \beta_{10} + \cdots + 7524 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1548 \beta_{16} - 1188 \beta_{14} - 6534 \beta_{13} + 6024 \beta_{11} + 11965 \beta_{10} + 17778 \beta_{8} + \cdots + 673596 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 40287 \beta_{17} - 274638 \beta_{15} + 289962 \beta_{12} + 625482 \beta_{11} + 938223 \beta_{10} + \cdots + 160056 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 20475 \beta_{16} - 27027 \beta_{14} - 108108 \beta_{13} - 3 \beta_{12} + 84609 \beta_{11} + \cdots + 7505784 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 593460 \beta_{17} - 4052160 \beta_{15} + 3666060 \beta_{12} + 7299580 \beta_{11} + 10947064 \beta_{10} + \cdots + 2800980 ) / 4 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 3 \beta_{17} + 241920 \beta_{16} - 1139 \beta_{15} - 498960 \beta_{14} - 1621620 \beta_{13} + \cdots + 84440070 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 8307900 \beta_{17} - 9208 \beta_{16} - 56816856 \beta_{15} - 4532 \beta_{14} + 45584208 \beta_{12} + \cdots + 43953840 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2252\mathbb{Z}\right)^\times\).

\(n\) \(565\) \(1127\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1125.1
−0.258155
0.545459
0.657449
−0.938912
1.42408
−1.68258
−1.78106
2.02273
−2.41825
2.61675
2.68985
−2.86258
3.12074
−3.23531
−3.27420
3.35716
−3.44682
3.46363
0 −5.93336 0 0 0 −10.1001 0 26.2047 0
1125.2 0 −5.70247 0 0 0 −6.39067 0 23.5182 0
1125.3 0 −5.56776 0 0 0 1.74496 0 22.0000 0
1125.4 0 −5.11845 0 0 0 6.17524 0 17.1985 0
1125.5 0 −3.97198 0 0 0 12.8068 0 6.77665 0
1125.6 0 −3.16894 0 0 0 10.2656 0 1.04216 0
1125.7 0 −2.82783 0 0 0 3.11122 0 −1.00338 0
1125.8 0 −1.90856 0 0 0 −1.50552 0 −5.35741 0
1125.9 0 −0.152076 0 0 0 −13.9688 0 −8.97687 0
1125.10 0 0.847382 0 0 0 −12.9023 0 −8.28194 0
1125.11 0 1.23527 0 0 0 −7.59213 0 −7.47410 0
1125.12 0 2.19437 0 0 0 −3.34579 0 −4.18476 0
1125.13 0 3.73899 0 0 0 13.4459 0 4.98004 0
1125.14 0 4.46720 0 0 0 13.9828 0 10.9559 0
1125.15 0 4.72038 0 0 0 11.1573 0 13.2820 0
1125.16 0 5.27052 0 0 0 7.79355 0 18.7784 0
1125.17 0 5.88054 0 0 0 −11.3013 0 25.5808 0
1125.18 0 5.99677 0 0 0 −13.3768 0 26.9612 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1125.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
563.b odd 2 1 CM by \(\Q(\sqrt{-563}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2252.3.d.a ✓ 18
563.b odd 2 1 CM 2252.3.d.a ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2252.3.d.a ✓ 18 1.a even 1 1 trivial
2252.3.d.a ✓ 18 563.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} - 162 T_{3}^{16} + 10935 T_{3}^{14} - 398034 T_{3}^{12} + 8444007 T_{3}^{10} + \cdots - 335296418 \) acting on \(S_{3}^{\mathrm{new}}(2252, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots - 335296418 \) Copy content Toggle raw display
$5$ \( T^{18} \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots - 48\!\cdots\!18 \) Copy content Toggle raw display
$11$ \( (T^{6} - 726 T^{4} + \cdots - 908882)^{3} \) Copy content Toggle raw display
$13$ \( (T^{6} - 1014 T^{4} + \cdots - 14027498)^{3} \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 61\!\cdots\!22 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 10\!\cdots\!38 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 93\!\cdots\!98 \) Copy content Toggle raw display
$29$ \( T^{18} \) Copy content Toggle raw display
$31$ \( T^{18} \) Copy content Toggle raw display
$37$ \( T^{18} \) Copy content Toggle raw display
$41$ \( T^{18} \) Copy content Toggle raw display
$43$ \( T^{18} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 37\!\cdots\!98 \) Copy content Toggle raw display
$53$ \( T^{18} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 68\!\cdots\!78 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 72\!\cdots\!98 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots - 19\!\cdots\!98 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 39\!\cdots\!82 \) Copy content Toggle raw display
$73$ \( T^{18} \) Copy content Toggle raw display
$79$ \( T^{18} \) Copy content Toggle raw display
$83$ \( T^{18} \) Copy content Toggle raw display
$89$ \( T^{18} \) Copy content Toggle raw display
$97$ \( T^{18} \) Copy content Toggle raw display
show more
show less