Properties

Label 294.3.b.c
Level $294$
Weight $3$
Character orbit 294.b
Analytic conductor $8.011$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [294,3,Mod(197,294)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(294, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("294.197");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 294.b (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: \(8.01091977219\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + (2 \beta + 1) q^{3} - 2 q^{4} + 6 \beta q^{5} + (\beta - 4) q^{6} - 2 \beta q^{8} + (4 \beta - 7) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + (2 \beta + 1) q^{3} - 2 q^{4} + 6 \beta q^{5} + (\beta - 4) q^{6} - 2 \beta q^{8} + (4 \beta - 7) q^{9} - 12 q^{10} + ( - 4 \beta - 2) q^{12} + q^{13} + (6 \beta - 24) q^{15} + 4 q^{16} - 6 \beta q^{17} + ( - 7 \beta - 8) q^{18} + 31 q^{19} - 12 \beta q^{20} + 6 \beta q^{23} + ( - 2 \beta + 8) q^{24} - 47 q^{25} + \beta q^{26} + ( - 10 \beta - 23) q^{27} - 12 \beta q^{29} + ( - 24 \beta - 12) q^{30} + 7 q^{31} + 4 \beta q^{32} + 12 q^{34} + ( - 8 \beta + 14) q^{36} - q^{37} + 31 \beta q^{38} + (2 \beta + 1) q^{39} + 24 q^{40} + 24 \beta q^{41} - 31 q^{43} + ( - 42 \beta - 48) q^{45} - 12 q^{46} + 30 \beta q^{47} + (8 \beta + 4) q^{48} - 47 \beta q^{50} + ( - 6 \beta + 24) q^{51} - 2 q^{52} - 18 \beta q^{53} + ( - 23 \beta + 20) q^{54} + (62 \beta + 31) q^{57} + 24 q^{58} - 6 \beta q^{59} + ( - 12 \beta + 48) q^{60} - 50 q^{61} + 7 \beta q^{62} - 8 q^{64} + 6 \beta q^{65} + 65 q^{67} + 12 \beta q^{68} + (6 \beta - 24) q^{69} + 42 \beta q^{71} + (14 \beta + 16) q^{72} + 97 q^{73} - \beta q^{74} + ( - 94 \beta - 47) q^{75} - 62 q^{76} + (\beta - 4) q^{78} - 103 q^{79} + 24 \beta q^{80} + ( - 56 \beta + 17) q^{81} - 48 q^{82} + 30 \beta q^{83} + 72 q^{85} - 31 \beta q^{86} + ( - 12 \beta + 48) q^{87} + 84 \beta q^{89} + ( - 48 \beta + 84) q^{90} - 12 \beta q^{92} + (14 \beta + 7) q^{93} - 60 q^{94} + 186 \beta q^{95} + (4 \beta - 16) q^{96} + 166 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 4 q^{4} - 8 q^{6} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 4 q^{4} - 8 q^{6} - 14 q^{9} - 24 q^{10} - 4 q^{12} + 2 q^{13} - 48 q^{15} + 8 q^{16} - 16 q^{18} + 62 q^{19} + 16 q^{24} - 94 q^{25} - 46 q^{27} - 24 q^{30} + 14 q^{31} + 24 q^{34} + 28 q^{36} - 2 q^{37} + 2 q^{39} + 48 q^{40} - 62 q^{43} - 96 q^{45} - 24 q^{46} + 8 q^{48} + 48 q^{51} - 4 q^{52} + 40 q^{54} + 62 q^{57} + 48 q^{58} + 96 q^{60} - 100 q^{61} - 16 q^{64} + 130 q^{67} - 48 q^{69} + 32 q^{72} + 194 q^{73} - 94 q^{75} - 124 q^{76} - 8 q^{78} - 206 q^{79} + 34 q^{81} - 96 q^{82} + 144 q^{85} + 96 q^{87} + 168 q^{90} + 14 q^{93} - 120 q^{94} - 32 q^{96} + 332 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(199\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
1.41421i
1.41421i
1.41421i 1.00000 2.82843i −2.00000 8.48528i −4.00000 1.41421i 0 2.82843i −7.00000 5.65685i −12.0000
197.2 1.41421i 1.00000 + 2.82843i −2.00000 8.48528i −4.00000 + 1.41421i 0 2.82843i −7.00000 + 5.65685i −12.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 294.3.b.c 2
3.b odd 2 1 inner 294.3.b.c 2
7.b odd 2 1 294.3.b.b 2
7.c even 3 2 294.3.h.b 4
7.d odd 6 2 42.3.h.a 4
21.c even 2 1 294.3.b.b 2
21.g even 6 2 42.3.h.a 4
21.h odd 6 2 294.3.h.b 4
28.f even 6 2 336.3.bn.c 4
84.j odd 6 2 336.3.bn.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.h.a 4 7.d odd 6 2
42.3.h.a 4 21.g even 6 2
294.3.b.b 2 7.b odd 2 1
294.3.b.b 2 21.c even 2 1
294.3.b.c 2 1.a even 1 1 trivial
294.3.b.c 2 3.b odd 2 1 inner
294.3.h.b 4 7.c even 3 2
294.3.h.b 4 21.h odd 6 2
336.3.bn.c 4 28.f even 6 2
336.3.bn.c 4 84.j odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(294, [\chi])\):

\( T_{5}^{2} + 72 \) Copy content Toggle raw display
\( T_{13} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2 \) Copy content Toggle raw display
$3$ \( T^{2} - 2T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + 72 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 72 \) Copy content Toggle raw display
$19$ \( (T - 31)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 72 \) Copy content Toggle raw display
$29$ \( T^{2} + 288 \) Copy content Toggle raw display
$31$ \( (T - 7)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 1152 \) Copy content Toggle raw display
$43$ \( (T + 31)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 1800 \) Copy content Toggle raw display
$53$ \( T^{2} + 648 \) Copy content Toggle raw display
$59$ \( T^{2} + 72 \) Copy content Toggle raw display
$61$ \( (T + 50)^{2} \) Copy content Toggle raw display
$67$ \( (T - 65)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 3528 \) Copy content Toggle raw display
$73$ \( (T - 97)^{2} \) Copy content Toggle raw display
$79$ \( (T + 103)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1800 \) Copy content Toggle raw display
$89$ \( T^{2} + 14112 \) Copy content Toggle raw display
$97$ \( (T - 166)^{2} \) Copy content Toggle raw display
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