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Properties

Label 455.2.j.g
Level $455$
Weight $2$
Character orbit 455.j
Analytic conductor $3.633$
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] = [455,2,Mod(261,455)] 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("455.261"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(455, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 455 = 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 455.j (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,1,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.63319329197\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
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: \( x^{20} - x^{19} + 17 x^{18} - 12 x^{17} + 181 x^{16} - 114 x^{15} + 1154 x^{14} - 605 x^{13} + \cdots + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_1) q^{2} - \beta_{12} q^{3} + (\beta_{15} - \beta_{9} - 1) q^{4} - \beta_{9} q^{5} + (\beta_{19} - \beta_{17} + \cdots - \beta_1) q^{6} + ( - \beta_{10} - \beta_{5}) q^{7} + ( - \beta_{19} + \beta_{17} - \beta_{16} + \cdots - 1) q^{8}+ \cdots + ( - 3 \beta_{16} + \beta_{14} + \cdots - 3 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + q^{2} + 4 q^{3} - 13 q^{4} + 10 q^{5} - 18 q^{6} + q^{7} - 6 q^{8} - 14 q^{9} - q^{10} - 7 q^{11} + 7 q^{12} - 20 q^{13} + 3 q^{14} + 8 q^{15} - 7 q^{16} + 6 q^{17} + 5 q^{18} + 10 q^{19} - 26 q^{20}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} + 17 x^{18} - 12 x^{17} + 181 x^{16} - 114 x^{15} + 1154 x^{14} - 605 x^{13} + \cdots + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 60\!\cdots\!54 \nu^{19} + \cdots - 17\!\cdots\!01 ) / 59\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 61\!\cdots\!26 \nu^{19} + \cdots - 17\!\cdots\!50 ) / 59\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 69\!\cdots\!49 \nu^{19} + \cdots - 19\!\cdots\!00 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\!\cdots\!39 \nu^{19} + \cdots + 16\!\cdots\!50 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 23\!\cdots\!52 \nu^{19} + \cdots + 10\!\cdots\!75 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 35\!\cdots\!46 \nu^{19} + \cdots + 19\!\cdots\!01 ) / 78\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 27\!\cdots\!11 \nu^{19} + \cdots + 23\!\cdots\!25 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 71\!\cdots\!38 \nu^{19} + \cdots - 64\!\cdots\!25 ) / 14\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 32\!\cdots\!09 \nu^{19} + \cdots + 58\!\cdots\!25 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 50\!\cdots\!98 \nu^{19} + \cdots - 14\!\cdots\!61 ) / 78\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 52\!\cdots\!17 \nu^{19} + \cdots - 79\!\cdots\!50 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 73\!\cdots\!87 \nu^{19} + \cdots + 67\!\cdots\!50 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 73\!\cdots\!11 \nu^{19} + \cdots - 13\!\cdots\!25 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 19\!\cdots\!64 \nu^{19} + \cdots + 24\!\cdots\!50 ) / 14\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 81\!\cdots\!59 \nu^{19} + \cdots + 39\!\cdots\!50 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 12\!\cdots\!04 \nu^{19} + \cdots - 10\!\cdots\!25 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 12\!\cdots\!98 \nu^{19} + \cdots - 55\!\cdots\!25 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 89\!\cdots\!89 \nu^{19} + \cdots - 14\!\cdots\!50 ) / 29\!\cdots\!50 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{15} + 3\beta_{9} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{19} + \beta_{17} - \beta_{16} + \beta_{15} - \beta_{12} - \beta_{11} - \beta_{10} - \beta_{6} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{18} + \beta_{17} + 7\beta_{15} + \beta_{14} + \beta_{13} - \beta_{10} - 14\beta_{9} - 2\beta_{6} - \beta_{4} - 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{19} + 10 \beta_{16} - 8 \beta_{15} + 9 \beta_{13} + 2 \beta_{12} + 9 \beta_{11} + \cdots - 29 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13 \beta_{19} - 13 \beta_{17} + 24 \beta_{16} - 13 \beta_{15} - 12 \beta_{14} + 13 \beta_{12} + \cdots + 76 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2 \beta_{18} - 80 \beta_{17} + 13 \beta_{16} - 27 \beta_{15} - 26 \beta_{14} - 71 \beta_{13} + \cdots + 73 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 126 \beta_{19} - 76 \beta_{18} - 191 \beta_{16} - 176 \beta_{15} - 126 \beta_{13} - 99 \beta_{12} + \cdots + 144 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 603 \beta_{19} + 603 \beta_{17} - 755 \beta_{16} + 603 \beta_{15} + 247 \beta_{14} - 603 \beta_{12} + \cdots - 576 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 530 \beta_{18} + 1095 \beta_{17} - 279 \beta_{16} + 2010 \beta_{15} + 878 \beta_{14} + 1090 \beta_{13} + \cdots - 2853 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4457 \beta_{19} + 370 \beta_{18} + 4758 \beta_{16} - 2039 \beta_{15} + 4046 \beta_{13} + \cdots - 7695 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 8997 \beta_{19} - 8997 \beta_{17} + 14347 \beta_{16} - 8997 \beta_{15} - 6824 \beta_{14} + 8997 \beta_{12} + \cdots + 18913 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 3528 \beta_{18} - 32746 \beta_{17} + 8203 \beta_{16} - 20274 \beta_{15} - 16786 \beta_{14} + \cdots + 34507 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 71508 \beta_{19} - 24419 \beta_{18} - 90872 \beta_{16} - 22445 \beta_{15} - 70343 \beta_{13} + \cdots + 90015 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 240411 \beta_{19} + 240411 \beta_{17} - 327054 \beta_{16} + 240411 \beta_{15} + 130854 \beta_{14} + \cdots - 263849 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 167210 \beta_{18} + 556510 \beta_{17} - 158097 \beta_{16} + 658772 \beta_{15} + 390324 \beta_{14} + \cdots - 910710 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 1767333 \beta_{19} + 256314 \beta_{18} + 1958630 \beta_{16} - 476592 \beta_{15} + 1656477 \beta_{13} + \cdots - 2613095 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 4271553 \beta_{19} - 4271553 \beta_{17} + 6389304 \beta_{16} - 4271553 \beta_{15} - 2919166 \beta_{14} + \cdots + 6504963 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 2059683 \beta_{18} - 13017668 \beta_{17} + 3612912 \beta_{16} - 10009289 \beta_{15} - 7613259 \beta_{14} + \cdots + 15178572 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/455\mathbb{Z}\right)^\times\).

\(n\) \(66\) \(92\) \(106\)
\(\chi(n)\) \(\beta_{9}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
261.1
−1.16947 − 2.02558i
−1.16558 − 2.01884i
−0.924791 − 1.60179i
−0.457041 − 0.791618i
−0.142795 − 0.247329i
0.337219 + 0.584081i
0.633675 + 1.09756i
0.930124 + 1.61102i
1.09447 + 1.89568i
1.36418 + 2.36284i
−1.16947 + 2.02558i
−1.16558 + 2.01884i
−0.924791 + 1.60179i
−0.457041 + 0.791618i
−0.142795 + 0.247329i
0.337219 − 0.584081i
0.633675 − 1.09756i
0.930124 − 1.61102i
1.09447 − 1.89568i
1.36418 − 2.36284i
−1.16947 + 2.02558i 0.630850 + 1.09266i −1.73531 − 3.00564i 0.500000 − 0.866025i −2.95103 −0.846187 − 2.50678i 3.43967 0.704056 − 1.21946i 1.16947 + 2.02558i
261.2 −1.16558 + 2.01884i 1.60462 + 2.77928i −1.71715 − 2.97419i 0.500000 − 0.866025i −7.48124 1.52339 + 2.16317i 3.34356 −3.64960 + 6.32129i 1.16558 + 2.01884i
261.3 −0.924791 + 1.60179i −0.676890 − 1.17241i −0.710478 − 1.23058i 0.500000 − 0.866025i 2.50393 −1.42021 + 2.23226i −1.07099 0.583640 − 1.01089i 0.924791 + 1.60179i
261.4 −0.457041 + 0.791618i −0.633151 − 1.09665i 0.582227 + 1.00845i 0.500000 − 0.866025i 1.15750 1.99561 − 1.73711i −2.89257 0.698240 − 1.20939i 0.457041 + 0.791618i
261.5 −0.142795 + 0.247329i 1.20822 + 2.09270i 0.959219 + 1.66142i 0.500000 − 0.866025i −0.690114 −2.20970 + 1.45507i −1.11907 −1.41960 + 2.45882i 0.142795 + 0.247329i
261.6 0.337219 − 0.584081i 0.578900 + 1.00268i 0.772566 + 1.33812i 0.500000 − 0.866025i 0.780865 0.136410 − 2.64223i 2.39097 0.829751 − 1.43717i −0.337219 − 0.584081i
261.7 0.633675 − 1.09756i −0.868620 − 1.50449i 0.196912 + 0.341062i 0.500000 − 0.866025i −2.20169 2.54129 + 0.736100i 3.03381 −0.00900122 + 0.0155906i −0.633675 − 1.09756i
261.8 0.930124 − 1.61102i 1.50544 + 2.60750i −0.730260 − 1.26485i 0.500000 − 0.866025i 5.60098 1.39315 − 2.24925i 1.00357 −3.03269 + 5.25277i −0.930124 − 1.61102i
261.9 1.09447 − 1.89568i −1.52380 − 2.63930i −1.39574 − 2.41749i 0.500000 − 0.866025i −6.67103 −2.53112 + 0.770335i −1.73250 −3.14394 + 5.44546i −1.09447 − 1.89568i
261.10 1.36418 − 2.36284i 0.174433 + 0.302127i −2.72199 − 4.71463i 0.500000 − 0.866025i 0.951836 −0.0826303 + 2.64446i −9.39646 1.43915 − 2.49267i −1.36418 − 2.36284i
326.1 −1.16947 − 2.02558i 0.630850 − 1.09266i −1.73531 + 3.00564i 0.500000 + 0.866025i −2.95103 −0.846187 + 2.50678i 3.43967 0.704056 + 1.21946i 1.16947 − 2.02558i
326.2 −1.16558 − 2.01884i 1.60462 − 2.77928i −1.71715 + 2.97419i 0.500000 + 0.866025i −7.48124 1.52339 − 2.16317i 3.34356 −3.64960 − 6.32129i 1.16558 − 2.01884i
326.3 −0.924791 − 1.60179i −0.676890 + 1.17241i −0.710478 + 1.23058i 0.500000 + 0.866025i 2.50393 −1.42021 − 2.23226i −1.07099 0.583640 + 1.01089i 0.924791 − 1.60179i
326.4 −0.457041 − 0.791618i −0.633151 + 1.09665i 0.582227 − 1.00845i 0.500000 + 0.866025i 1.15750 1.99561 + 1.73711i −2.89257 0.698240 + 1.20939i 0.457041 − 0.791618i
326.5 −0.142795 − 0.247329i 1.20822 − 2.09270i 0.959219 − 1.66142i 0.500000 + 0.866025i −0.690114 −2.20970 − 1.45507i −1.11907 −1.41960 − 2.45882i 0.142795 − 0.247329i
326.6 0.337219 + 0.584081i 0.578900 − 1.00268i 0.772566 − 1.33812i 0.500000 + 0.866025i 0.780865 0.136410 + 2.64223i 2.39097 0.829751 + 1.43717i −0.337219 + 0.584081i
326.7 0.633675 + 1.09756i −0.868620 + 1.50449i 0.196912 − 0.341062i 0.500000 + 0.866025i −2.20169 2.54129 − 0.736100i 3.03381 −0.00900122 − 0.0155906i −0.633675 + 1.09756i
326.8 0.930124 + 1.61102i 1.50544 − 2.60750i −0.730260 + 1.26485i 0.500000 + 0.866025i 5.60098 1.39315 + 2.24925i 1.00357 −3.03269 − 5.25277i −0.930124 + 1.61102i
326.9 1.09447 + 1.89568i −1.52380 + 2.63930i −1.39574 + 2.41749i 0.500000 + 0.866025i −6.67103 −2.53112 − 0.770335i −1.73250 −3.14394 − 5.44546i −1.09447 + 1.89568i
326.10 1.36418 + 2.36284i 0.174433 − 0.302127i −2.72199 + 4.71463i 0.500000 + 0.866025i 0.951836 −0.0826303 − 2.64446i −9.39646 1.43915 + 2.49267i −1.36418 + 2.36284i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 261.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 455.2.j.g ✓ 20
7.c even 3 1 inner 455.2.j.g ✓ 20
7.c even 3 1 3185.2.a.bb 10
7.d odd 6 1 3185.2.a.bc 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
455.2.j.g ✓ 20 1.a even 1 1 trivial
455.2.j.g ✓ 20 7.c even 3 1 inner
3185.2.a.bb 10 7.c even 3 1
3185.2.a.bc 10 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(455, [\chi])\):

\( T_{2}^{20} - T_{2}^{19} + 17 T_{2}^{18} - 12 T_{2}^{17} + 181 T_{2}^{16} - 114 T_{2}^{15} + 1154 T_{2}^{14} + \cdots + 625 \) Copy content Toggle raw display
\( T_{3}^{20} - 4 T_{3}^{19} + 30 T_{3}^{18} - 72 T_{3}^{17} + 404 T_{3}^{16} - 817 T_{3}^{15} + \cdots + 11664 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - T^{19} + \cdots + 625 \) Copy content Toggle raw display
$3$ \( T^{20} - 4 T^{19} + \cdots + 11664 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 282475249 \) Copy content Toggle raw display
$11$ \( T^{20} + 7 T^{19} + \cdots + 2624400 \) Copy content Toggle raw display
$13$ \( (T + 1)^{20} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 1173199504 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 28876884624 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 1179109313424 \) Copy content Toggle raw display
$29$ \( (T^{10} - 3 T^{9} + \cdots + 36173)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 32108632936704 \) Copy content Toggle raw display
$37$ \( T^{20} - 2 T^{19} + \cdots + 200704 \) Copy content Toggle raw display
$41$ \( (T^{10} + 19 T^{9} + \cdots - 4527936)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + T^{9} + \cdots - 10172)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 25352473695376 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 467905249296 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 268368875553024 \) Copy content Toggle raw display
$61$ \( T^{20} - 14 T^{19} + \cdots + 48841 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 308297094508881 \) Copy content Toggle raw display
$71$ \( (T^{10} - 7 T^{9} + \cdots - 101239344)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{10} + 20 T^{9} + \cdots - 7309428)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 231418209054096 \) Copy content Toggle raw display
$97$ \( (T^{10} + 19 T^{9} + \cdots - 1404768736)^{2} \) Copy content Toggle raw display
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