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Properties

Label 296.2.i.c
Level $296$
Weight $2$
Character orbit 296.i
Analytic conductor $2.364$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [296,2,Mod(121,296)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(296, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("296.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 296 = 2^{3} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 296.i (of order \(3\), degree \(2\), 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: \(2.36357189983\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 9x^{8} + 4x^{7} + 54x^{6} + 6x^{5} + 98x^{4} - 8x^{3} + 148x^{2} - 24x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{9} q^{3} + (\beta_{9} + \beta_{6} + \beta_{2}) q^{5} + ( - \beta_{8} - \beta_{7} + 1) q^{7} + (\beta_{9} - \beta_{8} + \cdots - \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{3} + (\beta_{9} + \beta_{6} + \beta_{2}) q^{5} + ( - \beta_{8} - \beta_{7} + 1) q^{7} + (\beta_{9} - \beta_{8} + \cdots - \beta_{4}) q^{9}+ \cdots + (2 \beta_{9} - 2 \beta_{8} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{3} + 3 q^{5} + 3 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{3} + 3 q^{5} + 3 q^{7} - 6 q^{9} + 16 q^{11} + q^{13} + 13 q^{15} + 5 q^{17} - 3 q^{19} + 3 q^{21} - 12 q^{23} - 12 q^{25} + 26 q^{27} - 16 q^{31} + 4 q^{33} + 5 q^{35} - 12 q^{37} + 3 q^{39} + 9 q^{41} + 4 q^{43} - 64 q^{45} + 32 q^{47} - 18 q^{49} - 2 q^{51} + 5 q^{53} + 4 q^{55} - 9 q^{57} - 5 q^{59} + 15 q^{61} - 96 q^{63} + 23 q^{65} + q^{67} + 12 q^{69} - 17 q^{71} - 36 q^{73} + 40 q^{75} - 4 q^{77} + 21 q^{79} - 17 q^{81} - 15 q^{83} + 6 q^{85} - 44 q^{87} + 29 q^{89} - q^{91} - 4 q^{93} - 21 q^{95} - 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 9x^{8} + 4x^{7} + 54x^{6} + 6x^{5} + 98x^{4} - 8x^{3} + 148x^{2} - 24x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13 \nu^{9} - 605 \nu^{8} + 1210 \nu^{7} - 7028 \nu^{6} + 1815 \nu^{5} - 27830 \nu^{4} + \cdots + 5505 ) / 15127 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 40 \nu^{9} + 33 \nu^{8} - 66 \nu^{7} + 1012 \nu^{6} - 99 \nu^{5} + 1518 \nu^{4} - 15676 \nu^{3} + \cdots + 5004 ) / 15127 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 146 \nu^{9} + 1309 \nu^{8} - 2618 \nu^{7} + 17092 \nu^{6} - 3927 \nu^{5} + 60214 \nu^{4} + \cdots + 49383 ) / 15127 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 153 \nu^{9} + 360 \nu^{8} - 720 \nu^{7} + 235 \nu^{6} - 1080 \nu^{5} + 16560 \nu^{4} + \cdots + 50660 ) / 15127 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 863 \nu^{9} - 1395 \nu^{8} + 7112 \nu^{7} + 4762 \nu^{6} + 27956 \nu^{5} + 7143 \nu^{4} + \cdots - 11542 ) / 30254 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1251 \nu^{9} + 1291 \nu^{8} - 11226 \nu^{7} - 5070 \nu^{6} - 66542 \nu^{5} - 7605 \nu^{4} + \cdots + 29562 ) / 30254 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2432 \nu^{9} - 1142 \nu^{8} - 17165 \nu^{7} - 36462 \nu^{6} - 156488 \nu^{5} - 123845 \nu^{4} + \cdots - 3432 ) / 30254 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4505 \nu^{9} - 4117 \nu^{8} + 40649 \nu^{7} + 22134 \nu^{6} + 243578 \nu^{5} + 65616 \nu^{4} + \cdots + 2796 ) / 30254 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} + \beta_{8} - 7\beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - 3\beta_{3} + 2\beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{9} - 4\beta_{8} + 21\beta_{7} + 9\beta_{6} + 9\beta_{2} - 6\beta _1 - 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} - 11\beta_{8} + 40\beta_{7} + 23\beta_{6} - \beta_{5} - 11\beta_{4} + 23\beta_{3} - 23\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -11\beta_{5} - 35\beta_{4} + 60\beta_{3} - 79\beta_{2} + 160 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -16\beta_{9} + 104\beta_{8} - 409\beta_{7} - 223\beta_{6} - 223\beta_{2} + 197\beta _1 + 409 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -78\beta_{9} + 316\beta_{8} - 1363\beta_{7} - 705\beta_{6} + 78\beta_{5} + 316\beta_{4} - 565\beta_{3} + 565\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 176\beta_{5} + 954\beta_{4} - 1757\beta_{3} + 2073\beta_{2} - 3877 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/296\mathbb{Z}\right)^\times\).

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(-1 + \beta_{7}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
0.0825878 + 0.143046i
1.50928 + 2.61415i
−0.831426 − 1.44007i
−0.917643 − 1.58940i
0.657199 + 1.13830i
0.0825878 − 0.143046i
1.50928 − 2.61415i
−0.831426 + 1.44007i
−0.917643 + 1.58940i
0.657199 − 1.13830i
0 −1.64462 − 2.84856i 0 1.87297 + 3.24408i 0 2.03658 + 3.52745i 0 −3.90954 + 6.77153i 0
121.2 0 −0.603312 − 1.04497i 0 −0.717824 − 1.24331i 0 −0.0542045 − 0.0938850i 0 0.772029 − 1.33719i 0
121.3 0 −0.261675 − 0.453235i 0 1.19680 + 2.07292i 0 −2.55985 − 4.43380i 0 1.36305 − 2.36088i 0
121.4 0 0.788575 + 1.36585i 0 −1.73584 − 3.00655i 0 1.47954 + 2.56263i 0 0.256299 − 0.443924i 0
121.5 0 1.22103 + 2.11489i 0 0.883889 + 1.53094i 0 0.597948 + 1.03568i 0 −1.48184 + 2.56662i 0
137.1 0 −1.64462 + 2.84856i 0 1.87297 − 3.24408i 0 2.03658 − 3.52745i 0 −3.90954 − 6.77153i 0
137.2 0 −0.603312 + 1.04497i 0 −0.717824 + 1.24331i 0 −0.0542045 + 0.0938850i 0 0.772029 + 1.33719i 0
137.3 0 −0.261675 + 0.453235i 0 1.19680 − 2.07292i 0 −2.55985 + 4.43380i 0 1.36305 + 2.36088i 0
137.4 0 0.788575 − 1.36585i 0 −1.73584 + 3.00655i 0 1.47954 − 2.56263i 0 0.256299 + 0.443924i 0
137.5 0 1.22103 − 2.11489i 0 0.883889 − 1.53094i 0 0.597948 − 1.03568i 0 −1.48184 − 2.56662i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 296.2.i.c ✓ 10
3.b odd 2 1 2664.2.r.n 10
4.b odd 2 1 592.2.i.h 10
37.c even 3 1 inner 296.2.i.c ✓ 10
111.i odd 6 1 2664.2.r.n 10
148.i odd 6 1 592.2.i.h 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
296.2.i.c ✓ 10 1.a even 1 1 trivial
296.2.i.c ✓ 10 37.c even 3 1 inner
592.2.i.h 10 4.b odd 2 1
592.2.i.h 10 148.i odd 6 1
2664.2.r.n 10 3.b odd 2 1
2664.2.r.n 10 111.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + T_{3}^{9} + 11T_{3}^{8} - 2T_{3}^{7} + 88T_{3}^{6} + 16T_{3}^{5} + 168T_{3}^{4} + 96T_{3}^{3} + 288T_{3}^{2} + 128T_{3} + 64 \) acting on \(S_{2}^{\mathrm{new}}(296, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + T^{9} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{10} - 3 T^{9} + \cdots + 6241 \) Copy content Toggle raw display
$7$ \( T^{10} - 3 T^{9} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( (T^{5} - 8 T^{4} + \cdots - 128)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} - T^{9} + \cdots + 2304 \) Copy content Toggle raw display
$17$ \( (T^{2} - T + 1)^{5} \) Copy content Toggle raw display
$19$ \( T^{10} + 3 T^{9} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( (T^{5} + 6 T^{4} + \cdots + 1152)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} - 72 T^{3} + \cdots + 358)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 8 T^{4} + \cdots + 128)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 12 T^{9} + \cdots + 69343957 \) Copy content Toggle raw display
$41$ \( T^{10} - 9 T^{9} + \cdots + 8543929 \) Copy content Toggle raw display
$43$ \( (T^{5} - 2 T^{4} - 16 T^{3} + \cdots + 32)^{2} \) Copy content Toggle raw display
$47$ \( (T^{5} - 16 T^{4} + \cdots + 23424)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} - 5 T^{9} + \cdots + 2304 \) Copy content Toggle raw display
$59$ \( T^{10} + 5 T^{9} + \cdots + 147456 \) Copy content Toggle raw display
$61$ \( T^{10} - 15 T^{9} + \cdots + 40921609 \) Copy content Toggle raw display
$67$ \( T^{10} - T^{9} + \cdots + 20736 \) Copy content Toggle raw display
$71$ \( T^{10} + 17 T^{9} + \cdots + 40000 \) Copy content Toggle raw display
$73$ \( (T^{5} + 18 T^{4} + \cdots + 1184)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 12908595456 \) Copy content Toggle raw display
$83$ \( T^{10} + 15 T^{9} + \cdots + 45481536 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 1372480209 \) Copy content Toggle raw display
$97$ \( (T^{5} + 18 T^{4} + \cdots + 88282)^{2} \) Copy content Toggle raw display
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