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Properties

Label 297.2.f.c
Level $297$
Weight $2$
Character orbit 297.f
Analytic conductor $2.372$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

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: \( x^{16} + 8x^{14} + 35x^{12} + 108x^{10} + 589x^{8} + 792x^{6} + 465x^{4} + 22x^{2} + 121 \) 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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + (\beta_{14} - 2 \beta_{9} - 2 \beta_{7} + \cdots + 1) q^{4}+ \cdots + (\beta_{15} + \beta_{12} + \cdots - \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + (\beta_{14} - 2 \beta_{9} - 2 \beta_{7} + \cdots + 1) q^{4}+ \cdots + ( - 2 \beta_{15} + \beta_{13} + \cdots + 6 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{4} + 8 q^{7} - 12 q^{10} + 8 q^{13} - 2 q^{16} + 10 q^{19} - 24 q^{22} - 16 q^{25} - 30 q^{28} - 6 q^{31} + 32 q^{34} + 12 q^{37} - 40 q^{40} - 80 q^{43} - 12 q^{46} + 40 q^{49} + 12 q^{52} + 56 q^{55} - 50 q^{58} - 6 q^{61} + 6 q^{64} + 64 q^{67} + 74 q^{70} - 32 q^{73} + 52 q^{76} - 4 q^{79} + 4 q^{82} + 70 q^{85} + 56 q^{88} - 94 q^{91} - 46 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 8x^{14} + 35x^{12} + 108x^{10} + 589x^{8} + 792x^{6} + 465x^{4} + 22x^{2} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1453738 \nu^{14} - 14940196 \nu^{12} - 37494977 \nu^{10} - 5191303 \nu^{8} + \cdots - 108606212 ) / 5958914918 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6073790 \nu^{14} - 52420744 \nu^{12} - 253232003 \nu^{10} - 852502804 \nu^{8} + \cdots - 1757970885 ) / 5958914918 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6073790 \nu^{15} + 52420744 \nu^{13} + 253232003 \nu^{11} + 852502804 \nu^{9} + \cdots + 1757970885 \nu ) / 5958914918 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 7060725 \nu^{14} - 86908445 \nu^{12} - 446152434 \nu^{10} - 1505258830 \nu^{8} + \cdots + 1446696614 ) / 5958914918 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7515186 \nu^{14} + 57757144 \nu^{12} + 246926149 \nu^{10} + 696402412 \nu^{8} + \cdots - 14257295349 ) / 5958914918 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10237697 \nu^{14} - 59584826 \nu^{12} - 192000045 \nu^{10} - 420957076 \nu^{8} + \cdots + 4245079553 ) / 5958914918 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10693842 \nu^{15} + 89901292 \nu^{13} + 468969029 \nu^{11} + 1699814305 \nu^{9} + \cdots + 3407335558 \nu ) / 5958914918 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13601318 \nu^{14} - 119781684 \nu^{12} - 543958983 \nu^{10} - 1710196911 \nu^{8} + \cdots + 2334366936 ) / 5958914918 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 13601318 \nu^{15} - 119781684 \nu^{13} - 543958983 \nu^{11} - 1710196911 \nu^{9} + \cdots + 2334366936 \nu ) / 5958914918 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 27017671 \nu^{15} - 211510658 \nu^{13} - 958001908 \nu^{11} - 3134565880 \nu^{9} + \cdots - 23003014254 \nu ) / 5958914918 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 29912805 \nu^{15} + 231787254 \nu^{13} + 989191031 \nu^{11} + 2983656791 \nu^{9} + \cdots + 1137439314 \nu ) / 5958914918 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 61281032 \nu^{15} + 537982471 \nu^{13} + 2492072308 \nu^{11} + 8060218532 \nu^{9} + \cdots + 16887583768 \nu ) / 5958914918 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 67421665 \nu^{14} - 529042412 \nu^{12} - 2282858893 \nu^{10} - 6988188262 \nu^{8} + \cdots - 2173556605 ) / 5958914918 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 7323136 \nu^{15} - 55354894 \nu^{13} - 227822551 \nu^{11} - 659839659 \nu^{9} + \cdots + 1634380270 \nu ) / 541719538 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - 2\beta_{3} - \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{8} + 4\beta_{4} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{14} - 6\beta_{9} + 7\beta_{7} + \beta_{6} + 6\beta_{5} + 7\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} - 6\beta_{13} - 8\beta_{12} - 6\beta_{11} - 26\beta_{10} + \beta_{8} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 32\beta_{14} - 55\beta_{9} - 96\beta_{7} - 23\beta_{6} - 32\beta_{5} - 67\beta_{3} + 23\beta_{2} + 35 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 32\beta_{15} + 32\beta_{13} + 137\beta_{12} + 41\beta_{11} + 137\beta_{10} - 84\beta_{4} - 84\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -169\beta_{14} + 358\beta_{9} + 391\beta_{7} + 62\beta_{5} + 358\beta_{3} - 62\beta_{2} - 391 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -169\beta_{15} - 62\beta_{13} - 729\beta_{12} - 231\beta_{11} - 326\beta_{10} - 62\beta_{8} + 326\beta_{4} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 388\beta_{14} - 1028\beta_{9} - 640\beta_{7} + 510\beta_{6} - 388\beta_{3} + 388\beta_{2} + 2113 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 388\beta_{15} + 1926\beta_{12} + 898\beta_{11} + 898\beta_{8} - 1926\beta_{4} + 1983\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4161\beta_{9} - 2314\beta_{7} - 2314\beta_{6} - 2314\beta_{5} - 4994\beta_{3} - 2493\beta_{2} - 6475 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2314\beta_{13} + 11103\beta_{10} - 4807\beta_{8} + 9980\beta_{4} - 11103\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -13417\beta_{14} + 578\beta_{9} + 49585\beta_{7} + 13417\beta_{6} + 25890\beta_{5} + 49585\beta_{3} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 13417 \beta_{15} - 25890 \beta_{13} - 63002 \beta_{12} - 25890 \beta_{11} - 114204 \beta_{10} + \cdots + 63002 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(56\) \(244\)
\(\chi(n)\) \(1\) \(-\beta_{7}\)

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
−1.60453 + 1.16576i
−0.541760 + 0.393612i
0.541760 − 0.393612i
1.60453 − 1.16576i
−0.723396 − 2.22638i
−0.329643 − 1.01454i
0.329643 + 1.01454i
0.723396 + 2.22638i
−1.60453 − 1.16576i
−0.541760 − 0.393612i
0.541760 + 0.393612i
1.60453 + 1.16576i
−0.723396 + 2.22638i
−0.329643 + 1.01454i
0.329643 − 1.01454i
0.723396 − 2.22638i
−0.612877 − 1.88624i 0 −1.56426 + 1.13650i −0.925742 + 2.84914i 0 3.34004 − 2.42668i −0.106651 − 0.0774867i 0 5.94154
82.2 −0.206934 − 0.636878i 0 1.25524 − 0.911987i 0.854331 − 2.62936i 0 −1.22201 + 0.887841i −1.92410 − 1.39794i 0 −1.85137
82.3 0.206934 + 0.636878i 0 1.25524 − 0.911987i −0.854331 + 2.62936i 0 −1.22201 + 0.887841i 1.92410 + 1.39794i 0 −1.85137
82.4 0.612877 + 1.88624i 0 −1.56426 + 1.13650i 0.925742 − 2.84914i 0 3.34004 − 2.42668i 0.106651 + 0.0774867i 0 5.94154
136.1 −1.89387 − 1.37598i 0 1.07540 + 3.30976i 2.90288 − 2.10907i 0 0.355620 + 1.09448i 1.07069 − 3.29523i 0 −8.39974
136.2 −0.863017 − 0.627019i 0 −0.266388 − 0.819857i −0.993168 + 0.721579i 0 −0.473654 − 1.45776i −0.943455 + 2.90366i 0 1.30957
136.3 0.863017 + 0.627019i 0 −0.266388 − 0.819857i 0.993168 − 0.721579i 0 −0.473654 − 1.45776i 0.943455 − 2.90366i 0 1.30957
136.4 1.89387 + 1.37598i 0 1.07540 + 3.30976i −2.90288 + 2.10907i 0 0.355620 + 1.09448i −1.07069 + 3.29523i 0 −8.39974
163.1 −0.612877 + 1.88624i 0 −1.56426 − 1.13650i −0.925742 − 2.84914i 0 3.34004 + 2.42668i −0.106651 + 0.0774867i 0 5.94154
163.2 −0.206934 + 0.636878i 0 1.25524 + 0.911987i 0.854331 + 2.62936i 0 −1.22201 − 0.887841i −1.92410 + 1.39794i 0 −1.85137
163.3 0.206934 − 0.636878i 0 1.25524 + 0.911987i −0.854331 − 2.62936i 0 −1.22201 − 0.887841i 1.92410 − 1.39794i 0 −1.85137
163.4 0.612877 − 1.88624i 0 −1.56426 − 1.13650i 0.925742 + 2.84914i 0 3.34004 + 2.42668i 0.106651 − 0.0774867i 0 5.94154
190.1 −1.89387 + 1.37598i 0 1.07540 − 3.30976i 2.90288 + 2.10907i 0 0.355620 − 1.09448i 1.07069 + 3.29523i 0 −8.39974
190.2 −0.863017 + 0.627019i 0 −0.266388 + 0.819857i −0.993168 − 0.721579i 0 −0.473654 + 1.45776i −0.943455 − 2.90366i 0 1.30957
190.3 0.863017 − 0.627019i 0 −0.266388 + 0.819857i 0.993168 + 0.721579i 0 −0.473654 + 1.45776i 0.943455 + 2.90366i 0 1.30957
190.4 1.89387 − 1.37598i 0 1.07540 − 3.30976i −2.90288 − 2.10907i 0 0.355620 − 1.09448i −1.07069 − 3.29523i 0 −8.39974
\(n\): 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
3.b odd 2 1 inner
11.c even 5 1 inner
33.h odd 10 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 3T_{2}^{14} + 25T_{2}^{12} + 143T_{2}^{10} + 494T_{2}^{8} + 167T_{2}^{6} + 575T_{2}^{4} + 407T_{2}^{2} + 121 \) acting on \(S_{2}^{\mathrm{new}}(297, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 3 T^{14} + \cdots + 121 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 18 T^{14} + \cdots + 1771561 \) Copy content Toggle raw display
$7$ \( (T^{8} - 4 T^{7} + \cdots + 121)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( (T^{8} - 4 T^{7} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 5208653241 \) Copy content Toggle raw display
$19$ \( (T^{8} - 5 T^{7} + 17 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 36 T^{6} + \cdots + 891)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 11623211721 \) Copy content Toggle raw display
$31$ \( (T^{8} + 3 T^{7} + \cdots + 1234321)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 6 T^{7} + \cdots + 30976)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 111746041 \) Copy content Toggle raw display
$43$ \( (T^{4} + 20 T^{3} + \cdots - 1251)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + 99 T^{14} + \cdots + 1771561 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 7502181258121 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 25937424601 \) Copy content Toggle raw display
$61$ \( (T^{8} + 3 T^{7} + \cdots + 9801)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 16 T^{3} + \cdots - 13761)^{4} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 536358317866921 \) Copy content Toggle raw display
$73$ \( (T^{8} + 16 T^{7} + \cdots + 2027776)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 2 T^{7} + \cdots + 3481)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 18463134765625 \) Copy content Toggle raw display
$89$ \( (T^{8} - 409 T^{6} + \cdots + 56133891)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 47 T^{6} + \cdots + 68121)^{2} \) Copy content Toggle raw display
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