↑

Properties

Label 735.2.b.f
Level $735$
Weight $2$
Character orbit 735.b
Analytic conductor $5.869$
Analytic rank $0$
Dimension $16$
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] = [735,2,Mod(146,735)] 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("735.146"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(735, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,4] 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: \(5.86900454856\)
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: \( x^{16} + 24x^{14} + 220x^{12} + 968x^{10} + 2133x^{8} + 2288x^{6} + 1052x^{4} + 128x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{7} q^{3} + (\beta_{14} - \beta_{10} + \cdots - \beta_{3}) q^{4} - q^{5} + (\beta_{14} + \beta_{13} + \cdots - \beta_{3}) q^{6} + ( - \beta_{14} - \beta_{13} + \cdots - \beta_1) q^{8}+ \cdots + (\beta_{13} - 2 \beta_{12} + 3 \beta_{7} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{3} - 16 q^{4} - 16 q^{5} - 4 q^{9} - 16 q^{12} - 4 q^{15} + 48 q^{16} - 24 q^{17} + 16 q^{20} + 32 q^{22} + 40 q^{24} + 16 q^{25} + 16 q^{26} + 16 q^{27} - 36 q^{33} + 16 q^{36} - 8 q^{38} - 12 q^{39}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 24x^{14} + 220x^{12} + 968x^{10} + 2133x^{8} + 2288x^{6} + 1052x^{4} + 128x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 109 \nu^{15} - 246 \nu^{14} + 2856 \nu^{13} - 5593 \nu^{12} + 29454 \nu^{11} - 46980 \nu^{10} + \cdots + 5314 ) / 1904 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{14} - 112\nu^{12} - 918\nu^{10} - 3304\nu^{8} - 4785\nu^{6} - 1432\nu^{4} + 1094\nu^{2} + 96 ) / 28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 251 \nu^{15} - 706 \nu^{14} + 6664 \nu^{13} - 16184 \nu^{12} + 69738 \nu^{11} - 137708 \nu^{10} + \cdots + 160 ) / 3808 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 74 \nu^{15} - 1547 \nu^{13} - 11044 \nu^{11} - 27202 \nu^{9} + 13294 \nu^{7} + 124125 \nu^{5} + \cdots + 13038 \nu ) / 952 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 251 \nu^{15} + 402 \nu^{14} + 6664 \nu^{13} + 9044 \nu^{12} + 69738 \nu^{11} + 74636 \nu^{10} + \cdots + 4232 ) / 3808 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 535 \nu^{15} + 768 \nu^{14} - 12852 \nu^{13} + 17850 \nu^{12} - 117938 \nu^{11} + 155632 \nu^{10} + \cdots + 15836 ) / 3808 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 71 \nu^{14} - 1649 \nu^{12} - 14362 \nu^{10} - 58022 \nu^{8} - 109871 \nu^{6} - 89705 \nu^{4} + \cdots - 1318 ) / 136 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 535 \nu^{15} + 768 \nu^{14} + 12852 \nu^{13} + 17850 \nu^{12} + 117938 \nu^{11} + 155632 \nu^{10} + \cdots + 15836 ) / 3808 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 359 \nu^{15} + 1888 \nu^{14} - 7378 \nu^{13} + 43316 \nu^{12} - 50658 \nu^{11} + 369664 \nu^{10} + \cdots + 10152 ) / 3808 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 591 \nu^{15} + 14280 \nu^{13} + 132162 \nu^{11} + 589400 \nu^{9} + 1320271 \nu^{7} + 1425376 \nu^{5} + \cdots + 34192 \nu ) / 1904 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 713 \nu^{15} + 894 \nu^{14} + 16184 \nu^{13} + 20230 \nu^{12} + 135678 \nu^{11} + 168596 \nu^{10} + \cdots - 2588 ) / 3808 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 327 \nu^{15} + 7735 \nu^{13} + 69322 \nu^{11} + 293818 \nu^{9} + 606407 \nu^{7} + 579247 \nu^{5} + \cdots + 12490 \nu ) / 952 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 359 \nu^{15} - 1888 \nu^{14} - 7378 \nu^{13} - 43316 \nu^{12} - 50658 \nu^{11} - 369664 \nu^{10} + \cdots - 10152 ) / 3808 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1825 \nu^{15} - 650 \nu^{14} - 42126 \nu^{13} - 14994 \nu^{12} - 363182 \nu^{11} - 129196 \nu^{10} + \cdots - 16332 ) / 3808 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} - \beta_{10} - \beta_{8} + \beta_{6} - \beta_{4} - \beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} - \beta_{13} + \beta_{11} - \beta_{10} + 2\beta_{5} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{15} - 6 \beta_{14} - 2 \beta_{12} - \beta_{11} + 9 \beta_{10} - 2 \beta_{9} + 8 \beta_{8} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{14} + 9 \beta_{13} + 2 \beta_{12} - 9 \beta_{11} + 10 \beta_{10} - \beta_{9} + \beta_{7} + \cdots + 31 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 32 \beta_{15} + 37 \beta_{14} + 24 \beta_{12} + 8 \beta_{11} - 69 \beta_{10} + 18 \beta_{9} - 61 \beta_{8} + \cdots - 88 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 81 \beta_{14} - 65 \beta_{13} - 23 \beta_{12} + 65 \beta_{11} - 81 \beta_{10} + 12 \beta_{9} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 274 \beta_{15} - 239 \beta_{14} - 218 \beta_{12} - 56 \beta_{11} + 513 \beta_{10} - 132 \beta_{9} + \cdots + 661 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 616 \beta_{14} + 442 \beta_{13} + 201 \beta_{12} - 447 \beta_{11} + 616 \beta_{10} - 103 \beta_{9} + \cdots + 17 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2174 \beta_{15} + 1604 \beta_{14} + 1793 \beta_{12} + 381 \beta_{11} - 3778 \beta_{10} + 909 \beta_{9} + \cdots - 4950 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4561 \beta_{14} - 2953 \beta_{13} - 1576 \beta_{12} + 3041 \beta_{11} - 4561 \beta_{10} + 783 \beta_{9} + \cdots - 217 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 16625 \beta_{15} - 11080 \beta_{14} - 14074 \beta_{12} - 2551 \beta_{11} + 27705 \beta_{10} + \cdots + 36880 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 33310 \beta_{14} + 19627 \beta_{13} + 11664 \beta_{12} - 20669 \beta_{11} + 33310 \beta_{10} + \cdots + 2410 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 124548 \beta_{15} + 78157 \beta_{14} + 107722 \beta_{12} + 16826 \beta_{11} - 202705 \beta_{10} + \cdots - 273720 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 241363 \beta_{14} - 130139 \beta_{13} - 83297 \beta_{12} + 140627 \beta_{11} - 241363 \beta_{10} + \cdots - 24425 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
146.1
− 2.72098i
− 2.66161i
− 2.19422i
− 1.51836i
− 1.07096i
− 1.04075i
− 0.336241i
− 0.221175i
0.221175i
0.336241i
1.04075i
1.07096i
1.51836i
2.19422i
2.66161i
2.72098i
− 2.72098i −0.193527 − 1.72121i −5.40376 −1.00000 −4.68337 + 0.526585i 0 9.26157i −2.92509 + 0.666201i 2.72098i
146.2 − 2.66161i 1.40479 + 1.01320i −5.08416 −1.00000 2.69674 − 3.73899i 0 8.20884i 0.946850 + 2.84666i 2.66161i
146.3 − 2.19422i 1.37239 − 1.05667i −2.81458 −1.00000 −2.31855 − 3.01132i 0 1.78737i 0.766915 − 2.90032i 2.19422i
146.4 − 1.51836i −1.50141 + 0.863574i −0.305408 −1.00000 1.31121 + 2.27968i 0 − 2.57300i 1.50848 − 2.59316i 1.51836i
146.5 − 1.07096i 0.759619 + 1.55659i 0.853043 −1.00000 1.66705 − 0.813522i 0 − 3.05550i −1.84596 + 2.36483i 1.07096i
146.6 − 1.04075i −0.633288 + 1.61212i 0.916843 −1.00000 1.67782 + 0.659093i 0 − 3.03570i −2.19789 − 2.04188i 1.04075i
146.7 − 0.336241i 1.72991 − 0.0860082i 1.88694 −1.00000 −0.0289195 − 0.581669i 0 − 1.30695i 2.98521 − 0.297574i 0.336241i
146.8 − 0.221175i −0.938482 − 1.45576i 1.95108 −1.00000 −0.321979 + 0.207569i 0 − 0.873880i −1.23850 + 2.73242i 0.221175i
146.9 0.221175i −0.938482 + 1.45576i 1.95108 −1.00000 −0.321979 − 0.207569i 0 0.873880i −1.23850 − 2.73242i − 0.221175i
146.10 0.336241i 1.72991 + 0.0860082i 1.88694 −1.00000 −0.0289195 + 0.581669i 0 1.30695i 2.98521 + 0.297574i − 0.336241i
146.11 1.04075i −0.633288 − 1.61212i 0.916843 −1.00000 1.67782 − 0.659093i 0 3.03570i −2.19789 + 2.04188i − 1.04075i
146.12 1.07096i 0.759619 − 1.55659i 0.853043 −1.00000 1.66705 + 0.813522i 0 3.05550i −1.84596 − 2.36483i − 1.07096i
146.13 1.51836i −1.50141 − 0.863574i −0.305408 −1.00000 1.31121 − 2.27968i 0 2.57300i 1.50848 + 2.59316i − 1.51836i
146.14 2.19422i 1.37239 + 1.05667i −2.81458 −1.00000 −2.31855 + 3.01132i 0 − 1.78737i 0.766915 + 2.90032i − 2.19422i
146.15 2.66161i 1.40479 − 1.01320i −5.08416 −1.00000 2.69674 + 3.73899i 0 − 8.20884i 0.946850 − 2.84666i − 2.66161i
146.16 2.72098i −0.193527 + 1.72121i −5.40376 −1.00000 −4.68337 − 0.526585i 0 − 9.26157i −2.92509 − 0.666201i − 2.72098i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 146.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.b.f yes 16
3.b odd 2 1 735.2.b.e ✓ 16
7.b odd 2 1 735.2.b.e ✓ 16
7.c even 3 2 735.2.s.m 32
7.d odd 6 2 735.2.s.n 32
21.c even 2 1 inner 735.2.b.f yes 16
21.g even 6 2 735.2.s.m 32
21.h odd 6 2 735.2.s.n 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.2.b.e ✓ 16 3.b odd 2 1
735.2.b.e ✓ 16 7.b odd 2 1
735.2.b.f yes 16 1.a even 1 1 trivial
735.2.b.f yes 16 21.c even 2 1 inner
735.2.s.m 32 7.c even 3 2
735.2.s.m 32 21.g even 6 2
735.2.s.n 32 7.d odd 6 2
735.2.s.n 32 21.h odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\):

\( T_{2}^{16} + 24T_{2}^{14} + 220T_{2}^{12} + 968T_{2}^{10} + 2133T_{2}^{8} + 2288T_{2}^{6} + 1052T_{2}^{4} + 128T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{17}^{8} + 12T_{17}^{7} + 6T_{17}^{6} - 332T_{17}^{5} - 877T_{17}^{4} + 720T_{17}^{3} + 2716T_{17}^{2} + 48T_{17} - 1636 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 24 T^{14} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{16} - 4 T^{15} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( (T + 1)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + 108 T^{14} + \cdots + 21381376 \) Copy content Toggle raw display
$13$ \( T^{16} + 140 T^{14} + \cdots + 66846976 \) Copy content Toggle raw display
$17$ \( (T^{8} + 12 T^{7} + \cdots - 1636)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 101929216 \) Copy content Toggle raw display
$23$ \( T^{16} + 152 T^{14} + \cdots + 2446096 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 255825547264 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 1194920079376 \) Copy content Toggle raw display
$37$ \( (T^{8} - 168 T^{6} + \cdots + 286432)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 8 T^{7} + \cdots + 28448)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 8 T^{7} + \cdots + 6881536)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 4 T^{7} + \cdots - 1134532)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 361456144 \) Copy content Toggle raw display
$59$ \( (T^{8} + 8 T^{7} + \cdots + 2708608)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 10912727296 \) Copy content Toggle raw display
$67$ \( (T^{8} - 24 T^{7} + \cdots + 28544)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 7948079104 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 9751957504 \) Copy content Toggle raw display
$79$ \( (T^{8} + 20 T^{7} + \cdots + 608548)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 48 T^{7} + \cdots - 50176)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 16 T^{7} + \cdots + 7224832)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 542426358016 \) Copy content Toggle raw display
show more
show less