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Properties

Label 297.2.f.d
Level $297$
Weight $2$
Character orbit 297.f
Analytic conductor $2.372$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [297,2,Mod(82,297)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(297, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("297.82");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 297.f (of order \(5\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.37155694003\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 8 x^{14} - 22 x^{13} + 62 x^{12} - 24 x^{11} + 152 x^{10} - 161 x^{9} + 552 x^{8} + \cdots + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + ( - \beta_{12} + \beta_{6}) q^{2} + ( - \beta_{14} - \beta_{13} - \beta_{12} + \cdots - 1) q^{4}+ \cdots + ( - \beta_{14} - 2 \beta_{13} + \cdots - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{12} + \beta_{6}) q^{2} + ( - \beta_{14} - \beta_{13} - \beta_{12} + \cdots - 1) q^{4}+ \cdots + (3 \beta_{15} + 2 \beta_{14} + \cdots + 7) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{2} - 4 q^{4} + q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{2} - 4 q^{4} + q^{5} - 2 q^{7} + 6 q^{10} + 13 q^{11} - 2 q^{13} - 22 q^{14} - 24 q^{16} - 2 q^{17} - 2 q^{19} + 15 q^{22} + 14 q^{23} - 19 q^{25} + 21 q^{26} + 15 q^{28} + q^{29} + 14 q^{31} - 48 q^{32} + 10 q^{34} - 18 q^{35} + 9 q^{37} + 11 q^{38} + 33 q^{40} + 25 q^{41} + 14 q^{43} + 14 q^{44} + 4 q^{46} - 28 q^{47} - 4 q^{49} - 63 q^{50} + 10 q^{52} + q^{53} - 40 q^{55} + 96 q^{56} - 20 q^{58} + 41 q^{59} - 5 q^{62} - 92 q^{64} - 60 q^{65} - 48 q^{67} + 25 q^{68} - 31 q^{70} + 3 q^{71} - 13 q^{73} + 29 q^{74} - 58 q^{76} - 2 q^{77} - 83 q^{80} + 41 q^{82} - 14 q^{83} - 10 q^{85} - 56 q^{86} + 86 q^{88} + 82 q^{89} + 14 q^{91} + 74 q^{92} - 2 q^{94} - 56 q^{95} + 12 q^{97} + 26 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 8 x^{14} - 22 x^{13} + 62 x^{12} - 24 x^{11} + 152 x^{10} - 161 x^{9} + 552 x^{8} + \cdots + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 94\!\cdots\!82 \nu^{15} + \cdots + 14\!\cdots\!25 ) / 31\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 29\!\cdots\!08 \nu^{15} + \cdots + 10\!\cdots\!30 ) / 31\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 51\!\cdots\!83 \nu^{15} + \cdots - 38\!\cdots\!30 ) / 31\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 56\!\cdots\!57 \nu^{15} + \cdots - 72\!\cdots\!60 ) / 31\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 64\!\cdots\!66 \nu^{15} + \cdots + 11\!\cdots\!75 ) / 31\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 82\!\cdots\!98 \nu^{15} + \cdots - 85\!\cdots\!40 ) / 31\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11\!\cdots\!88 \nu^{15} + \cdots - 25\!\cdots\!40 ) / 31\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15\!\cdots\!46 \nu^{15} + \cdots + 63\!\cdots\!05 ) / 31\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 19\!\cdots\!56 \nu^{15} + \cdots + 23\!\cdots\!55 ) / 31\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 23\!\cdots\!81 \nu^{15} + \cdots + 12\!\cdots\!50 ) / 31\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 27\!\cdots\!43 \nu^{15} + \cdots - 42\!\cdots\!05 ) / 31\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 45\!\cdots\!80 \nu^{15} + \cdots + 22\!\cdots\!75 ) / 31\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 46\!\cdots\!20 \nu^{15} + \cdots + 71\!\cdots\!60 ) / 31\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 11\!\cdots\!76 \nu^{15} + \cdots - 17\!\cdots\!00 ) / 31\!\cdots\!55 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} - 2\beta_{11} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + 2 \beta_{14} + 2 \beta_{13} + 2 \beta_{12} - \beta_{11} + 3 \beta_{10} - \beta_{8} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{15} + 8\beta_{14} + 8\beta_{12} + \beta_{11} + 9\beta_{6} - \beta_{5} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{14} - 8 \beta_{12} - 9 \beta_{10} - 28 \beta_{7} + 6 \beta_{6} - 17 \beta_{5} - 25 \beta_{4} + \cdots - 71 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 22 \beta_{13} + \beta_{12} + 4 \beta_{11} + 62 \beta_{10} + 19 \beta_{9} - \beta_{8} + 23 \beta_{7} + \cdots + 30 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3 \beta_{15} + 77 \beta_{14} - 90 \beta_{13} + 4 \beta_{12} + 144 \beta_{11} + 3 \beta_{10} + 58 \beta_{9} + \cdots + 110 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 148 \beta_{15} - 434 \beta_{14} - 434 \beta_{13} - 437 \beta_{12} + 273 \beta_{11} - 633 \beta_{10} + \cdots - 1568 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 418 \beta_{15} - 1153 \beta_{14} + 190 \beta_{13} - 638 \beta_{12} - 500 \beta_{11} + \cdots - 309 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 190 \beta_{15} + 1893 \beta_{14} + 190 \beta_{13} + 1353 \beta_{12} + 190 \beta_{11} + 1893 \beta_{10} + \cdots + 12227 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 5013 \beta_{13} - 601 \beta_{12} + 1086 \beta_{11} - 10704 \beta_{10} - 3033 \beta_{9} + 601 \beta_{8} + \cdots - 9653 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1980 \beta_{15} - 16238 \beta_{14} + 14048 \beta_{13} - 3474 \beta_{12} - 23409 \beta_{11} + \cdots - 28764 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 22171 \beta_{15} + 66011 \beta_{14} + 66011 \beta_{13} + 63319 \beta_{12} - 35013 \beta_{11} + \cdots + 268309 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 59876 \beta_{15} + 182281 \beta_{14} - 58201 \beta_{13} + 96163 \beta_{12} + 97408 \beta_{11} + \cdots + 80063 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 58201 \beta_{15} - 397893 \beta_{14} - 58201 \beta_{13} - 282531 \beta_{12} - 58201 \beta_{11} + \cdots - 2112858 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/297\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(244\)
\(\chi(n)\) \(1\) \(-1 - \beta_{5} + \beta_{11} - \beta_{12}\)

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.

comment: embeddings in the coefficient field
 
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
0.416955 − 1.28326i
0.231171 − 0.711471i
−0.391187 + 1.20395i
−0.874974 + 2.69289i
2.13024 − 1.54771i
1.10209 − 0.800713i
−0.175229 + 0.127311i
−1.43906 + 1.04554i
0.416955 + 1.28326i
0.231171 + 0.711471i
−0.391187 − 1.20395i
−0.874974 − 2.69289i
2.13024 + 1.54771i
1.10209 + 0.800713i
−0.175229 − 0.127311i
−1.43906 − 1.04554i
−0.725972 − 2.23431i 0 −2.84709 + 2.06853i 1.06108 − 3.26568i 0 2.30657 − 1.67582i 2.88741 + 2.09782i 0 −8.06687
82.2 −0.540188 − 1.66253i 0 −0.854162 + 0.620585i −1.19305 + 3.67184i 0 −0.710219 + 0.516004i −1.33531 − 0.970162i 0 6.74902
82.3 0.0821696 + 0.252892i 0 1.56083 − 1.13401i 0.340091 − 1.04669i 0 1.24902 − 0.907465i 0.845280 + 0.614132i 0 0.292645
82.4 0.565957 + 1.74184i 0 −1.09565 + 0.796038i −0.517138 + 1.59159i 0 −3.34537 + 2.43055i 0.956730 + 0.695105i 0 −3.06496
136.1 −1.32122 − 0.959922i 0 0.206136 + 0.634423i −2.42257 + 1.76010i 0 −0.469421 − 1.44473i −0.672677 + 2.07029i 0 4.89031
136.2 −0.293070 − 0.212928i 0 −0.577482 − 1.77731i 2.17280 − 1.57863i 0 −0.770730 − 2.37206i −0.433081 + 1.33288i 0 −0.972914
136.3 0.984246 + 0.715096i 0 −0.160657 − 0.494452i 0.240251 − 0.174552i 0 1.32806 + 4.08735i 0.947351 − 2.91565i 0 0.361288
136.4 2.24808 + 1.63332i 0 1.76807 + 5.44156i 0.818541 − 0.594705i 0 −0.587909 − 1.80940i −3.19570 + 9.83534i 0 2.81149
163.1 −0.725972 + 2.23431i 0 −2.84709 − 2.06853i 1.06108 + 3.26568i 0 2.30657 + 1.67582i 2.88741 − 2.09782i 0 −8.06687
163.2 −0.540188 + 1.66253i 0 −0.854162 − 0.620585i −1.19305 − 3.67184i 0 −0.710219 − 0.516004i −1.33531 + 0.970162i 0 6.74902
163.3 0.0821696 − 0.252892i 0 1.56083 + 1.13401i 0.340091 + 1.04669i 0 1.24902 + 0.907465i 0.845280 − 0.614132i 0 0.292645
163.4 0.565957 − 1.74184i 0 −1.09565 − 0.796038i −0.517138 − 1.59159i 0 −3.34537 − 2.43055i 0.956730 − 0.695105i 0 −3.06496
190.1 −1.32122 + 0.959922i 0 0.206136 − 0.634423i −2.42257 − 1.76010i 0 −0.469421 + 1.44473i −0.672677 − 2.07029i 0 4.89031
190.2 −0.293070 + 0.212928i 0 −0.577482 + 1.77731i 2.17280 + 1.57863i 0 −0.770730 + 2.37206i −0.433081 − 1.33288i 0 −0.972914
190.3 0.984246 − 0.715096i 0 −0.160657 + 0.494452i 0.240251 + 0.174552i 0 1.32806 − 4.08735i 0.947351 + 2.91565i 0 0.361288
190.4 2.24808 − 1.63332i 0 1.76807 − 5.44156i 0.818541 + 0.594705i 0 −0.587909 + 1.80940i −3.19570 − 9.83534i 0 2.81149
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 297.2.f.d yes 16
3.b odd 2 1 297.2.f.a ✓ 16
9.c even 3 2 891.2.n.f 32
9.d odd 6 2 891.2.n.i 32
11.c even 5 1 inner 297.2.f.d yes 16
11.c even 5 1 3267.2.a.be 8
11.d odd 10 1 3267.2.a.bl 8
33.f even 10 1 3267.2.a.bf 8
33.h odd 10 1 297.2.f.a ✓ 16
33.h odd 10 1 3267.2.a.bm 8
99.m even 15 2 891.2.n.f 32
99.n odd 30 2 891.2.n.i 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.2.f.a ✓ 16 3.b odd 2 1
297.2.f.a ✓ 16 33.h odd 10 1
297.2.f.d yes 16 1.a even 1 1 trivial
297.2.f.d yes 16 11.c even 5 1 inner
891.2.n.f 32 9.c even 3 2
891.2.n.f 32 99.m even 15 2
891.2.n.i 32 9.d odd 6 2
891.2.n.i 32 99.n odd 30 2
3267.2.a.be 8 11.c even 5 1
3267.2.a.bf 8 33.f even 10 1
3267.2.a.bl 8 11.d odd 10 1
3267.2.a.bm 8 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 2 T_{2}^{15} + 8 T_{2}^{14} - 16 T_{2}^{13} + 56 T_{2}^{12} - 16 T_{2}^{11} + 202 T_{2}^{10} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(297, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 2 T^{15} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - T^{15} + \cdots + 3481 \) Copy content Toggle raw display
$7$ \( T^{16} + 2 T^{15} + \cdots + 245025 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( T^{16} + 2 T^{15} + \cdots + 12033961 \) Copy content Toggle raw display
$17$ \( T^{16} + 2 T^{15} + \cdots + 301401 \) Copy content Toggle raw display
$19$ \( T^{16} + 2 T^{15} + \cdots + 2418025 \) Copy content Toggle raw display
$23$ \( (T^{8} - 7 T^{7} + \cdots - 4779)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 598829841 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 29333755441 \) Copy content Toggle raw display
$37$ \( T^{16} - 9 T^{15} + \cdots + 4946176 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 1906282921 \) Copy content Toggle raw display
$43$ \( (T^{8} - 7 T^{7} + \cdots - 7524)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 6662278537321 \) Copy content Toggle raw display
$53$ \( T^{16} - T^{15} + \cdots + 657721 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 235375374025 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 9151026921 \) Copy content Toggle raw display
$67$ \( (T^{8} + 24 T^{7} + \cdots - 383139)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 14231297025 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 1761510273961 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 19833840390400 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 239452656921 \) Copy content Toggle raw display
$89$ \( (T^{8} - 41 T^{7} + \cdots + 2011959)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 226876931856 \) Copy content Toggle raw display
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