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Properties

Label 495.3.b.c
Level $495$
Weight $3$
Character orbit 495.b
Analytic conductor $13.488$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [495,3,Mod(406,495)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(495, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("495.406"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 495.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.4877730858\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 36x^{14} + 500x^{12} + 3364x^{10} + 11310x^{8} + 17932x^{6} + 12708x^{4} + 3244x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} + \beta_{4} q^{5} + (\beta_{15} - \beta_{13} + \cdots + \beta_{7}) q^{7} + (\beta_{8} - \beta_{7} - \beta_1) q^{8} + (\beta_{7} + \beta_1) q^{10} + ( - \beta_{15} - \beta_{13} - \beta_{12} + \cdots + 2) q^{11}+ \cdots + ( - 14 \beta_{15} + 6 \beta_{12} + \cdots - 56 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{4} + 28 q^{11} - 16 q^{16} - 40 q^{20} - 20 q^{22} - 56 q^{23} + 80 q^{25} + 88 q^{26} - 96 q^{31} - 200 q^{34} + 184 q^{37} - 296 q^{38} - 300 q^{44} + 200 q^{47} - 496 q^{49} + 80 q^{53} + 20 q^{55}+ \cdots - 144 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 36x^{14} + 500x^{12} + 3364x^{10} + 11310x^{8} + 17932x^{6} + 12708x^{4} + 3244x^{2} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 35 \nu^{14} + 1223 \nu^{12} + 16291 \nu^{10} + 102783 \nu^{8} + 308249 \nu^{6} + 377925 \nu^{4} + \cdots - 15435 ) / 5872 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 62 \nu^{14} + 2135 \nu^{12} + 27642 \nu^{10} + 164635 \nu^{8} + 433802 \nu^{6} + 369733 \nu^{4} + \cdots - 12295 ) / 5872 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{14} + 35\nu^{12} + 465\nu^{10} + 2899\nu^{8} + 8419\nu^{6} + 9657\nu^{4} + 3755\nu^{2} + 273 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 184 \nu^{15} - 4532 \nu^{14} + 6052 \nu^{13} - 158015 \nu^{12} + 71782 \nu^{11} - 2087998 \nu^{10} + \cdots - 1033175 ) / 64592 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 62 \nu^{15} + 2135 \nu^{13} + 27642 \nu^{11} + 164635 \nu^{9} + 433802 \nu^{7} + 369733 \nu^{5} + \cdots - 18167 \nu ) / 5872 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 62 \nu^{15} + 2135 \nu^{13} + 27642 \nu^{11} + 164635 \nu^{9} + 433802 \nu^{7} + 369733 \nu^{5} + \cdots + 34681 \nu ) / 5872 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1316 \nu^{15} + 47673 \nu^{13} + 668032 \nu^{11} + 4551885 \nu^{9} + 15564332 \nu^{7} + \cdots + 3194239 \nu ) / 64592 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3035 \nu^{15} + 1309 \nu^{14} - 105003 \nu^{13} + 47355 \nu^{12} - 1369671 \nu^{11} + \cdots + 1094049 ) / 129184 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3035 \nu^{15} - 4631 \nu^{14} + 105003 \nu^{13} - 161359 \nu^{12} + 1369671 \nu^{11} + \cdots + 411323 ) / 129184 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3035 \nu^{15} + 1309 \nu^{14} + 105003 \nu^{13} + 47355 \nu^{12} + 1369671 \nu^{11} + \cdots + 1094049 ) / 129184 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 2039 \nu^{15} - 70137 \nu^{13} - 905111 \nu^{11} - 5340393 \nu^{9} - 13619053 \nu^{7} + \cdots + 3474805 \nu ) / 64592 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2363 \nu^{15} - 82549 \nu^{13} - 1094171 \nu^{11} - 6803405 \nu^{9} - 19711721 \nu^{7} + \cdots - 451607 \nu ) / 64592 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2660 \nu^{15} - 92581 \nu^{13} - 1219032 \nu^{11} - 7483777 \nu^{9} - 21092804 \nu^{7} + \cdots - 1002883 \nu ) / 64592 \) 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
\(\nu^{3}\)\(=\) \( \beta_{8} - \beta_{7} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{10} + 3\beta_{4} - \beta_{3} - 10\beta_{2} + 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{14} + \beta_{9} - 12\beta_{8} + 17\beta_{7} + 88\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{12} - 15\beta_{11} - 17\beta_{10} + 2\beta_{5} - 53\beta_{4} + 15\beta_{3} + 99\beta_{2} - 392 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2\beta_{15} - 38\beta_{14} - 4\beta_{12} + 4\beta_{10} - 19\beta_{9} + 133\beta_{8} - 218\beta_{7} - 892\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 4 \beta_{13} + 36 \beta_{12} + 182 \beta_{11} + 226 \beta_{10} + 4 \beta_{8} + 4 \beta_{7} + \cdots + 3827 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 36 \beta_{15} + 538 \beta_{14} + 10 \beta_{13} + 104 \beta_{12} - 104 \beta_{10} + 262 \beta_{9} + \cdots + 9211 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 104 \beta_{13} - 492 \beta_{12} - 2034 \beta_{11} - 2734 \beta_{10} - 104 \beta_{8} - 104 \beta_{7} + \cdots - 38379 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 416 \beta_{15} - 6874 \beta_{14} - 254 \beta_{13} - 1804 \beta_{12} + 1804 \beta_{10} + \cdots - 96135 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1804 \beta_{13} + 6204 \beta_{12} + 21751 \beta_{11} + 31563 \beta_{10} + 1804 \beta_{8} + \cdots + 391572 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 3596 \beta_{15} + 83760 \beta_{14} + 4278 \beta_{13} + 26376 \beta_{12} - 26376 \beta_{10} + \cdots + 1009842 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 26376 \beta_{13} - 75886 \beta_{12} - 226465 \beta_{11} - 355103 \beta_{10} - 26376 \beta_{8} + \cdots - 4040102 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 19962 \beta_{15} - 994276 \beta_{14} - 60626 \beta_{13} - 352152 \beta_{12} + 352152 \beta_{10} + \cdots - 10650962 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/495\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(397\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
406.1
− 3.28542i
− 3.17407i
− 2.59216i
− 2.33103i
− 1.15335i
− 1.07765i
− 0.664903i
− 0.211240i
0.211240i
0.664903i
1.07765i
1.15335i
2.33103i
2.59216i
3.17407i
3.28542i
− 3.28542i 0 −6.79397 2.23607 0 10.0009i 9.17937i 0 − 7.34642i
406.2 − 3.17407i 0 −6.07473 2.23607 0 − 13.2254i 6.58534i 0 − 7.09744i
406.3 − 2.59216i 0 −2.71931 −2.23607 0 − 0.880263i − 3.31976i 0 5.79625i
406.4 − 2.33103i 0 −1.43371 −2.23607 0 6.32430i − 5.98210i 0 5.21235i
406.5 − 1.15335i 0 2.66978 −2.23607 0 − 12.5112i − 7.69260i 0 2.57897i
406.6 − 1.07765i 0 2.83866 2.23607 0 3.30417i − 7.36971i 0 − 2.40971i
406.7 − 0.664903i 0 3.55790 2.23607 0 8.36247i − 5.02527i 0 − 1.48677i
406.8 − 0.211240i 0 3.95538 −2.23607 0 9.32319i − 1.68049i 0 0.472346i
406.9 0.211240i 0 3.95538 −2.23607 0 − 9.32319i 1.68049i 0 − 0.472346i
406.10 0.664903i 0 3.55790 2.23607 0 − 8.36247i 5.02527i 0 1.48677i
406.11 1.07765i 0 2.83866 2.23607 0 − 3.30417i 7.36971i 0 2.40971i
406.12 1.15335i 0 2.66978 −2.23607 0 12.5112i 7.69260i 0 − 2.57897i
406.13 2.33103i 0 −1.43371 −2.23607 0 − 6.32430i 5.98210i 0 − 5.21235i
406.14 2.59216i 0 −2.71931 −2.23607 0 0.880263i 3.31976i 0 − 5.79625i
406.15 3.17407i 0 −6.07473 2.23607 0 13.2254i − 6.58534i 0 7.09744i
406.16 3.28542i 0 −6.79397 2.23607 0 − 10.0009i − 9.17937i 0 7.34642i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 406.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.3.b.c 16
3.b odd 2 1 165.3.b.a ✓ 16
11.b odd 2 1 inner 495.3.b.c 16
12.b even 2 1 2640.3.c.c 16
15.d odd 2 1 825.3.b.d 16
15.e even 4 2 825.3.h.b 32
33.d even 2 1 165.3.b.a ✓ 16
132.d odd 2 1 2640.3.c.c 16
165.d even 2 1 825.3.b.d 16
165.l odd 4 2 825.3.h.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.3.b.a ✓ 16 3.b odd 2 1
165.3.b.a ✓ 16 33.d even 2 1
495.3.b.c 16 1.a even 1 1 trivial
495.3.b.c 16 11.b odd 2 1 inner
825.3.b.d 16 15.d odd 2 1
825.3.b.d 16 165.d even 2 1
825.3.h.b 32 15.e even 4 2
825.3.h.b 32 165.l odd 4 2
2640.3.c.c 16 12.b even 2 1
2640.3.c.c 16 132.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 36T_{2}^{14} + 500T_{2}^{12} + 3364T_{2}^{10} + 11310T_{2}^{8} + 17932T_{2}^{6} + 12708T_{2}^{4} + 3244T_{2}^{2} + 121 \) acting on \(S_{3}^{\mathrm{new}}(495, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 36 T^{14} + \cdots + 121 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 5632002297856 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 45\!\cdots\!61 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 343146874802176 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 16781934923776 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( (T^{8} + 28 T^{7} + \cdots + 1312605184)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{8} + 48 T^{7} + \cdots - 282734392064)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 1511893588736)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 26\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots - 7749548297216)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 40 T^{7} + \cdots + 280885168384)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 68 T^{7} + \cdots + 232610439424)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 38480434282496)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 2604723965696)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 38\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 36\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 70\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 53\!\cdots\!84)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 54466210388224)^{2} \) Copy content Toggle raw display
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