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Properties

Label 405.3.g.h
Level $405$
Weight $3$
Character orbit 405.g
Analytic conductor $11.035$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [405,3,Mod(82,405)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([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("405.82"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 405.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.0354507066\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} + 2 x^{18} + 8 x^{17} + 245 x^{16} - 440 x^{15} + 422 x^{14} + 1724 x^{13} + \cdots + 11449 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( - \beta_{9} - 2 \beta_{5}) q^{4} - \beta_{12} q^{5} + (\beta_{10} - \beta_{3}) q^{7} + ( - \beta_{19} - \beta_{18} + \cdots - \beta_{4}) q^{8} + (\beta_{18} - \beta_{17} + \beta_{16} + \cdots - 1) q^{10}+ \cdots + (2 \beta_{17} - 2 \beta_{16} + \cdots - 47) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{2} + 2 q^{5} + 2 q^{7} - 12 q^{8} - 4 q^{10} - 8 q^{11} + 2 q^{13} - 28 q^{16} + 14 q^{17} + 114 q^{20} - 14 q^{22} - 82 q^{23} + 8 q^{25} - 56 q^{26} - 44 q^{28} + 4 q^{31} + 14 q^{32} + 176 q^{35}+ \cdots - 938 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} + 2 x^{18} + 8 x^{17} + 245 x^{16} - 440 x^{15} + 422 x^{14} + 1724 x^{13} + \cdots + 11449 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16\!\cdots\!77 \nu^{19} + \cdots + 34\!\cdots\!65 ) / 58\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 32\!\cdots\!87 \nu^{19} + \cdots + 19\!\cdots\!05 ) / 58\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\!\cdots\!15 \nu^{19} + \cdots - 12\!\cdots\!03 ) / 16\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17\!\cdots\!15 \nu^{19} + \cdots - 98\!\cdots\!17 ) / 62\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 64\!\cdots\!03 \nu^{19} + \cdots + 21\!\cdots\!25 ) / 16\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17\!\cdots\!93 \nu^{19} + \cdots - 10\!\cdots\!87 ) / 33\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 53\!\cdots\!71 \nu^{19} + \cdots + 58\!\cdots\!61 ) / 33\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 53\!\cdots\!45 \nu^{19} + \cdots + 29\!\cdots\!51 ) / 31\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 14\!\cdots\!03 \nu^{19} + \cdots - 13\!\cdots\!69 ) / 82\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 30\!\cdots\!07 \nu^{19} + \cdots - 26\!\cdots\!77 ) / 16\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 63\!\cdots\!57 \nu^{19} + \cdots + 36\!\cdots\!99 ) / 33\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 18\!\cdots\!82 \nu^{19} + \cdots + 37\!\cdots\!25 ) / 82\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 81\!\cdots\!27 \nu^{19} + \cdots - 16\!\cdots\!93 ) / 33\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 53\!\cdots\!73 \nu^{19} + \cdots - 31\!\cdots\!55 ) / 16\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 27\!\cdots\!34 \nu^{19} + \cdots + 44\!\cdots\!65 ) / 77\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 12\!\cdots\!55 \nu^{19} + \cdots - 23\!\cdots\!37 ) / 33\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 12\!\cdots\!69 \nu^{19} + \cdots - 72\!\cdots\!83 ) / 33\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 31\!\cdots\!50 \nu^{19} + \cdots + 51\!\cdots\!87 ) / 77\!\cdots\!16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 6\beta_{5} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{19} + \beta_{18} + \beta_{11} - \beta_{10} - \beta_{7} + \beta_{6} - 8\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} - \beta_{17} - \beta_{12} - \beta_{11} - \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} + 12\beta_{2} - \beta _1 - 58 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 16 \beta_{19} - 16 \beta_{18} + 15 \beta_{17} + \beta_{16} + \beta_{15} - \beta_{14} - 16 \beta_{13} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 19 \beta_{18} - 19 \beta_{17} + 3 \beta_{14} - \beta_{13} + 19 \beta_{12} - 19 \beta_{11} + \cdots - 9 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 215 \beta_{19} - 215 \beta_{18} - 21 \beta_{16} + 21 \beta_{15} - 25 \beta_{12} - 213 \beta_{11} + \cdots - 72 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 298 \beta_{19} + 289 \beta_{17} - 2 \beta_{16} - 71 \beta_{14} + 4 \beta_{13} + 265 \beta_{12} + \cdots + 7449 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2753 \beta_{19} + 2753 \beta_{18} - 2432 \beta_{17} - 328 \beta_{16} - 328 \beta_{15} + 312 \beta_{14} + \cdots - 1799 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 4430 \beta_{18} + 4111 \beta_{17} - 62 \beta_{15} - 1203 \beta_{14} - 240 \beta_{13} + \cdots - 1091 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 34680 \beta_{19} + 34680 \beta_{18} + 4626 \beta_{16} - 4626 \beta_{15} + 6872 \beta_{12} + \cdots + 33929 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 64021 \beta_{19} - 56752 \beta_{17} + 1182 \beta_{16} + 17985 \beta_{14} + 7706 \beta_{13} + \cdots - 1087328 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 434609 \beta_{19} - 434609 \beta_{18} + 378365 \beta_{17} + 62331 \beta_{16} + 62331 \beta_{15} + \cdots + 563964 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 906571 \beta_{18} - 770014 \beta_{17} + 17942 \beta_{15} + 253244 \beta_{14} + 159443 \beta_{13} + \cdots + 565222 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 5442616 \beta_{19} - 5442616 \beta_{18} - 819985 \beta_{16} + 819985 \beta_{15} - 1421803 \beta_{12} + \cdots - 8742929 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 12634924 \beta_{19} + 10327152 \beta_{17} - 235424 \beta_{16} - 3450444 \beta_{14} - 2777484 \beta_{13} + \cdots + 165137905 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 68238164 \beta_{19} + 68238164 \beta_{18} - 59675144 \beta_{17} - 10635368 \beta_{16} - 10635368 \beta_{15} + \cdots - 129775660 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 173880912 \beta_{18} + 137342904 \beta_{17} - 2732976 \beta_{15} - 46080240 \beta_{14} + \cdots - 145782540 \beta_1 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 857227785 \beta_{19} + 857227785 \beta_{18} + 136676628 \beta_{16} - 136676628 \beta_{15} + \cdots + 1870601388 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\) \(82\) \(326\)
\(\chi(n)\) \(-\beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1
−2.54217 + 2.54217i
−1.76130 + 1.76130i
−1.35293 + 1.35293i
−1.18732 + 1.18732i
0.134438 − 0.134438i
0.531225 − 0.531225i
0.652864 − 0.652864i
1.74749 − 1.74749i
2.36030 − 2.36030i
2.41740 − 2.41740i
−2.54217 − 2.54217i
−1.76130 − 1.76130i
−1.35293 − 1.35293i
−1.18732 − 1.18732i
0.134438 + 0.134438i
0.531225 + 0.531225i
0.652864 + 0.652864i
1.74749 + 1.74749i
2.36030 + 2.36030i
2.41740 + 2.41740i
−2.54217 − 2.54217i 0 8.92522i −2.58043 − 4.28268i 0 3.50586 + 3.50586i 12.5207 − 12.5207i 0 −4.32742 + 17.4472i
82.2 −1.76130 − 1.76130i 0 2.20434i 4.13877 + 2.80545i 0 0.0481814 + 0.0481814i −3.16269 + 3.16269i 0 −2.34837 − 12.2309i
82.3 −1.35293 − 1.35293i 0 − 0.339176i 1.72567 − 4.69277i 0 2.20824 + 2.20824i −5.87059 + 5.87059i 0 −8.68368 + 4.01426i
82.4 −1.18732 − 1.18732i 0 − 1.18052i −2.02288 + 4.57252i 0 −1.94050 − 1.94050i −6.15096 + 6.15096i 0 7.83089 − 3.02725i
82.5 0.134438 + 0.134438i 0 − 3.96385i −3.28806 − 3.76678i 0 −6.37245 − 6.37245i 1.07064 − 1.07064i 0 0.0643587 − 0.948439i
82.6 0.531225 + 0.531225i 0 − 3.43560i −3.56385 + 3.50699i 0 1.31570 + 1.31570i 3.94998 − 3.94998i 0 −3.75621 − 0.0302083i
82.7 0.652864 + 0.652864i 0 − 3.14754i 4.83588 + 1.27052i 0 8.09166 + 8.09166i 4.66637 − 4.66637i 0 2.32770 + 3.98665i
82.8 1.74749 + 1.74749i 0 2.10741i 4.94625 + 0.731199i 0 −7.58568 − 7.58568i 3.30727 − 3.30727i 0 7.36573 + 9.92126i
82.9 2.36030 + 2.36030i 0 7.14205i 1.54169 − 4.75639i 0 6.44182 + 6.44182i −7.41620 + 7.41620i 0 14.8654 − 7.58766i
82.10 2.41740 + 2.41740i 0 7.68766i −4.73304 + 1.61194i 0 −4.71284 − 4.71284i −8.91454 + 8.91454i 0 −15.3384 − 7.54494i
163.1 −2.54217 + 2.54217i 0 − 8.92522i −2.58043 + 4.28268i 0 3.50586 − 3.50586i 12.5207 + 12.5207i 0 −4.32742 − 17.4472i
163.2 −1.76130 + 1.76130i 0 − 2.20434i 4.13877 − 2.80545i 0 0.0481814 − 0.0481814i −3.16269 − 3.16269i 0 −2.34837 + 12.2309i
163.3 −1.35293 + 1.35293i 0 0.339176i 1.72567 + 4.69277i 0 2.20824 − 2.20824i −5.87059 − 5.87059i 0 −8.68368 − 4.01426i
163.4 −1.18732 + 1.18732i 0 1.18052i −2.02288 − 4.57252i 0 −1.94050 + 1.94050i −6.15096 − 6.15096i 0 7.83089 + 3.02725i
163.5 0.134438 − 0.134438i 0 3.96385i −3.28806 + 3.76678i 0 −6.37245 + 6.37245i 1.07064 + 1.07064i 0 0.0643587 + 0.948439i
163.6 0.531225 − 0.531225i 0 3.43560i −3.56385 − 3.50699i 0 1.31570 − 1.31570i 3.94998 + 3.94998i 0 −3.75621 + 0.0302083i
163.7 0.652864 − 0.652864i 0 3.14754i 4.83588 − 1.27052i 0 8.09166 − 8.09166i 4.66637 + 4.66637i 0 2.32770 − 3.98665i
163.8 1.74749 − 1.74749i 0 − 2.10741i 4.94625 − 0.731199i 0 −7.58568 + 7.58568i 3.30727 + 3.30727i 0 7.36573 − 9.92126i
163.9 2.36030 − 2.36030i 0 − 7.14205i 1.54169 + 4.75639i 0 6.44182 − 6.44182i −7.41620 − 7.41620i 0 14.8654 + 7.58766i
163.10 2.41740 − 2.41740i 0 − 7.68766i −4.73304 − 1.61194i 0 −4.71284 + 4.71284i −8.91454 − 8.91454i 0 −15.3384 + 7.54494i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.3.g.h 20
3.b odd 2 1 405.3.g.g 20
5.c odd 4 1 inner 405.3.g.h 20
9.c even 3 2 45.3.k.a ✓ 40
9.d odd 6 2 135.3.l.a 40
15.e even 4 1 405.3.g.g 20
45.j even 6 2 225.3.o.b 40
45.k odd 12 2 45.3.k.a ✓ 40
45.k odd 12 2 225.3.o.b 40
45.l even 12 2 135.3.l.a 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.3.k.a ✓ 40 9.c even 3 2
45.3.k.a ✓ 40 45.k odd 12 2
135.3.l.a 40 9.d odd 6 2
135.3.l.a 40 45.l even 12 2
225.3.o.b 40 45.j even 6 2
225.3.o.b 40 45.k odd 12 2
405.3.g.g 20 3.b odd 2 1
405.3.g.g 20 15.e even 4 1
405.3.g.h 20 1.a even 1 1 trivial
405.3.g.h 20 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 2 T_{2}^{19} + 2 T_{2}^{18} + 8 T_{2}^{17} + 245 T_{2}^{16} - 440 T_{2}^{15} + 422 T_{2}^{14} + \cdots + 11449 \) acting on \(S_{3}^{\mathrm{new}}(405, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 2 T^{19} + \cdots + 11449 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 95367431640625 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 130961476996 \) Copy content Toggle raw display
$11$ \( (T^{10} + 4 T^{9} + \cdots + 347037556)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 36\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 66729987548560)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 26448633442405)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 33\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 46\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots - 34\!\cdots\!38)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 15\!\cdots\!01 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 59\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 32\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 96\!\cdots\!64 \) Copy content Toggle raw display
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