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Properties

Label 297.3.c.a
Level $297$
Weight $3$
Character orbit 297.c
Analytic conductor $8.093$
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] = [297,3,Mod(109,297)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(297, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("297.109");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 297.c (of order \(2\), degree \(1\), 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: \(8.09266385150\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 212 x^{14} + 20776 x^{12} - 1001288 x^{10} + 21274804 x^{8} - 80418176 x^{6} + \cdots + 421886622784 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{11} q^{2} + ( - \beta_{2} - 2) q^{4} + \beta_{10} q^{5} - \beta_{4} q^{7} + (\beta_{14} + 2 \beta_{11}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{11} q^{2} + ( - \beta_{2} - 2) q^{4} + \beta_{10} q^{5} - \beta_{4} q^{7} + (\beta_{14} + 2 \beta_{11}) q^{8} - \beta_{6} q^{10} + ( - \beta_{10} - \beta_{9}) q^{11} + \beta_{5} q^{13} + \beta_{12} q^{14} + (\beta_{3} + \beta_1 + 2) q^{16} + ( - \beta_{15} - \beta_{14} + \cdots - \beta_{9}) q^{17}+ \cdots + ( - 4 \beta_{15} - 11 \beta_{14} + \cdots - 4 \beta_{9}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{4} + 40 q^{16} + 24 q^{22} + 120 q^{25} - 88 q^{31} - 216 q^{34} + 56 q^{37} - 32 q^{49} - 280 q^{55} + 120 q^{58} + 376 q^{64} + 200 q^{67} - 192 q^{70} + 456 q^{82} - 168 q^{88} - 264 q^{91} + 80 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 212 x^{14} + 20776 x^{12} - 1001288 x^{10} + 21274804 x^{8} - 80418176 x^{6} + \cdots + 421886622784 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 5184982179 \nu^{14} + 2076134434504 \nu^{12} - 283639029738396 \nu^{10} + \cdots - 26\!\cdots\!88 ) / 33\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 42\!\cdots\!79 \nu^{14} + \cdots - 35\!\cdots\!20 ) / 61\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 58\!\cdots\!83 \nu^{14} + \cdots - 39\!\cdots\!28 ) / 61\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 14\!\cdots\!03 \nu^{14} + \cdots + 62\!\cdots\!92 ) / 12\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15\!\cdots\!49 \nu^{14} + \cdots - 20\!\cdots\!48 ) / 12\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11\!\cdots\!77 \nu^{14} + \cdots - 15\!\cdots\!32 ) / 61\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 99\!\cdots\!41 \nu^{14} + \cdots - 15\!\cdots\!48 ) / 12\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10\!\cdots\!57 \nu^{15} + \cdots - 82\!\cdots\!84 \nu ) / 20\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 67\!\cdots\!83 \nu^{15} + \cdots - 96\!\cdots\!52 \nu ) / 41\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 77\!\cdots\!55 \nu^{15} + \cdots + 26\!\cdots\!12 \nu ) / 41\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 77\!\cdots\!55 \nu^{15} + \cdots + 15\!\cdots\!16 \nu ) / 41\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 33\!\cdots\!93 \nu^{15} + \cdots + 56\!\cdots\!88 \nu ) / 83\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 19\!\cdots\!91 \nu^{15} + \cdots - 16\!\cdots\!32 \nu ) / 41\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 64\!\cdots\!83 \nu^{15} + \cdots + 45\!\cdots\!48 \nu ) / 83\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 81\!\cdots\!43 \nu^{15} + \cdots + 16\!\cdots\!96 \nu ) / 83\!\cdots\!56 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{11} + \beta_{10} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{6} + 3\beta_{3} - \beta_{2} + 2\beta _1 + 25 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 6\beta_{15} + 23\beta_{14} + 15\beta_{13} - \beta_{12} + 116\beta_{11} + 35\beta_{10} + 12\beta_{9} + \beta_{8} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -36\beta_{7} + 212\beta_{6} + 20\beta_{5} - 16\beta_{4} + 52\beta_{3} - 102\beta_{2} - 58\beta _1 + 398 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 1018 \beta_{15} + 2020 \beta_{14} + 1102 \beta_{13} - 146 \beta_{12} + 9440 \beta_{11} + \cdots + 388 \beta_{8} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2228 \beta_{7} + 12100 \beta_{6} + 1232 \beta_{5} - 4 \beta_{4} - 9006 \beta_{3} - 21918 \beta_{2} + \cdots - 80690 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 88700 \beta_{15} + 62682 \beta_{14} + 35738 \beta_{13} - 9982 \beta_{12} + 325828 \beta_{11} + \cdots + 52766 \beta_{8} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 33328 \beta_{7} + 54384 \beta_{6} + 91536 \beta_{5} + 133504 \beta_{4} - 1206164 \beta_{3} + \cdots - 10370128 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4231884 \beta_{15} - 7307124 \beta_{14} - 3173448 \beta_{13} - 607840 \beta_{12} + \cdots + 4027180 \beta_{8} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 21332128 \beta_{7} - 70711104 \beta_{6} + 7250056 \beta_{5} + 22663816 \beta_{4} - 71669196 \beta_{3} + \cdots - 609490988 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 55095872 \beta_{15} - 1315123348 \beta_{14} - 608750908 \beta_{13} - 16615052 \beta_{12} + \cdots + 128820468 \beta_{8} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2346868400 \beta_{7} - 8325025360 \beta_{6} + 521698480 \beta_{5} + 2147540512 \beta_{4} + \cdots + 360391384 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 31955278040 \beta_{15} - 105375071776 \beta_{14} - 49251370872 \beta_{13} + \cdots - 11743529440 \beta_{8} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 132890346640 \beta_{7} - 477690697072 \beta_{6} + 26316184928 \beta_{5} + 117530646256 \beta_{4} + \cdots + 4148802911688 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3190476244368 \beta_{15} - 3464853636776 \beta_{14} - 1618099370952 \beta_{13} + \cdots - 2223330647032 \beta_{8} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(56\) \(244\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−8.35855 + 3.47232i
8.35855 + 3.47232i
−0.983062 + 3.03358i
0.983062 + 3.03358i
−3.98989 + 1.29424i
3.98989 + 1.29424i
−6.57640 + 1.03215i
6.57640 + 1.03215i
−6.57640 − 1.03215i
6.57640 − 1.03215i
−3.98989 − 1.29424i
3.98989 − 1.29424i
−0.983062 − 3.03358i
0.983062 − 3.03358i
−8.35855 − 3.47232i
8.35855 − 3.47232i
− 3.47232i 0 −8.05699 −8.35855 0 0.555248i 14.0872i 0 29.0235i
109.2 − 3.47232i 0 −8.05699 8.35855 0 − 0.555248i 14.0872i 0 − 29.0235i
109.3 − 3.03358i 0 −5.20262 −0.983062 0 10.1575i 3.64824i 0 2.98220i
109.4 − 3.03358i 0 −5.20262 0.983062 0 − 10.1575i 3.64824i 0 − 2.98220i
109.5 − 1.29424i 0 2.32494 −3.98989 0 − 7.86477i − 8.18599i 0 5.16388i
109.6 − 1.29424i 0 2.32494 3.98989 0 7.86477i − 8.18599i 0 − 5.16388i
109.7 − 1.03215i 0 2.93467 −6.57640 0 6.21784i − 7.15761i 0 6.78782i
109.8 − 1.03215i 0 2.93467 6.57640 0 − 6.21784i − 7.15761i 0 − 6.78782i
109.9 1.03215i 0 2.93467 −6.57640 0 − 6.21784i 7.15761i 0 − 6.78782i
109.10 1.03215i 0 2.93467 6.57640 0 6.21784i 7.15761i 0 6.78782i
109.11 1.29424i 0 2.32494 −3.98989 0 7.86477i 8.18599i 0 − 5.16388i
109.12 1.29424i 0 2.32494 3.98989 0 − 7.86477i 8.18599i 0 5.16388i
109.13 3.03358i 0 −5.20262 −0.983062 0 − 10.1575i − 3.64824i 0 − 2.98220i
109.14 3.03358i 0 −5.20262 0.983062 0 10.1575i − 3.64824i 0 2.98220i
109.15 3.47232i 0 −8.05699 −8.35855 0 − 0.555248i − 14.0872i 0 − 29.0235i
109.16 3.47232i 0 −8.05699 8.35855 0 0.555248i − 14.0872i 0 29.0235i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 297.3.c.a ✓ 16
3.b odd 2 1 inner 297.3.c.a ✓ 16
11.b odd 2 1 inner 297.3.c.a ✓ 16
33.d even 2 1 inner 297.3.c.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.3.c.a ✓ 16 1.a even 1 1 trivial
297.3.c.a ✓ 16 3.b odd 2 1 inner
297.3.c.a ✓ 16 11.b odd 2 1 inner
297.3.c.a ✓ 16 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 24T_{2}^{6} + 171T_{2}^{4} + 342T_{2}^{2} + 198 \) acting on \(S_{3}^{\mathrm{new}}(297, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 24 T^{6} + \cdots + 198)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 130 T^{6} + \cdots + 46486)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 204 T^{6} + \cdots + 76068)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 45\!\cdots\!61 \) Copy content Toggle raw display
$13$ \( (T^{8} + 960 T^{6} + \cdots + 2301057)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 1260 T^{6} + \cdots + 2710422)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 1512 T^{6} + \cdots + 2982169872)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 1132 T^{6} + \cdots + 185944)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 3822 T^{6} + \cdots + 121503092832)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 22 T^{3} + \cdots - 48752)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 14 T^{3} + \cdots - 150128)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + 4254 T^{6} + \cdots + 166133088)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 9487522271232)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 97996922578456)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 101498114823094)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 67026898237024)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 7997353060833)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 50 T^{3} + \cdots + 2034469)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 466013567568736)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 58298808518208)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 12\!\cdots\!17)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 32\!\cdots\!52)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 228015749685856)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 20 T^{3} + \cdots - 1863347)^{4} \) Copy content Toggle raw display
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