Properties

Label 294.3.b.i
Level $294$
Weight $3$
Character orbit 294.b
Analytic conductor $8.011$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} + 11x^{2} - 10x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{3} + \beta_{2} + \beta_1) q^{3} - 2 q^{4} + ( - 2 \beta_{3} + 2 \beta_{2} + 1) q^{5} + ( - 2 \beta_{3} + \beta_{2} + \beta_1) q^{6} - 2 \beta_{2} q^{8} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{3} + \beta_{2} + \beta_1) q^{3} - 2 q^{4} + ( - 2 \beta_{3} + 2 \beta_{2} + 1) q^{5} + ( - 2 \beta_{3} + \beta_{2} + \beta_1) q^{6} - 2 \beta_{2} q^{8} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 3) q^{9} + ( - \beta_{2} - 2 \beta_1 - 4) q^{10} + ( - 10 \beta_{3} + \beta_{2} + 5) q^{11} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{12} + (2 \beta_{2} + 4 \beta_1 - 8) q^{13} + ( - 5 \beta_{3} + 7 \beta_{2} + \beta_1 + 6) q^{15} + 4 q^{16} + (2 \beta_{3} - 2 \beta_{2} - 1) q^{17} + ( - 4 \beta_{3} + 2 \beta_{2} - \beta_1 + 12) q^{18} + ( - 3 \beta_{2} - 6 \beta_1 - 1) q^{19} + (4 \beta_{3} - 4 \beta_{2} - 2) q^{20} + ( - 5 \beta_{2} - 10 \beta_1 - 2) q^{22} + ( - 10 \beta_{3} + 7 \beta_{2} + 5) q^{23} + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{24} + ( - 4 \beta_{2} - 8 \beta_1 + 6) q^{25} + ( - 8 \beta_{3} - 8 \beta_{2} + 4) q^{26} + (10 \beta_{3} - 5 \beta_{2} - 2 \beta_1 + 15) q^{27} + ( - 20 \beta_{3} - 10 \beta_{2} + 10) q^{29} + ( - 2 \beta_{3} + \beta_{2} - 5 \beta_1 - 12) q^{30} + (5 \beta_{2} + 10 \beta_1 - 5) q^{31} + 4 \beta_{2} q^{32} + ( - 7 \beta_{3} + 26 \beta_{2} - 4 \beta_1 + 30) q^{33} + (\beta_{2} + 2 \beta_1 + 4) q^{34} + (2 \beta_{3} + 8 \beta_{2} - 4 \beta_1 - 6) q^{36} - q^{37} + (12 \beta_{3} - \beta_{2} - 6) q^{38} + ( - 12 \beta_{3} - 18 \beta_{2} - 6 \beta_1 + 24) q^{39} + (2 \beta_{2} + 4 \beta_1 + 8) q^{40} + (12 \beta_{3} + 6 \beta_{2} - 6) q^{41} + (6 \beta_{2} + 12 \beta_1 + 26) q^{43} + (20 \beta_{3} - 2 \beta_{2} - 10) q^{44} + ( - 13 \beta_{3} + 20 \beta_{2} + 8 \beta_1 + 21) q^{45} + ( - 5 \beta_{2} - 10 \beta_1 - 14) q^{46} + ( - 6 \beta_{3} - 45 \beta_{2} + 3) q^{47} + (4 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{48} + (16 \beta_{3} + 6 \beta_{2} - 8) q^{50} + (5 \beta_{3} - 7 \beta_{2} - \beta_1 - 6) q^{51} + ( - 4 \beta_{2} - 8 \beta_1 + 16) q^{52} + (6 \beta_{3} + 36 \beta_{2} - 3) q^{53} + (4 \beta_{3} + 25 \beta_{2} + 10 \beta_1 + 6) q^{54} + ( - 11 \beta_{2} - 22 \beta_1 - 59) q^{55} + (5 \beta_{3} + 14 \beta_{2} - 4 \beta_1 - 36) q^{57} + ( - 10 \beta_{2} - 20 \beta_1 + 20) q^{58} + ( - 14 \beta_{3} - 55 \beta_{2} + 7) q^{59} + (10 \beta_{3} - 14 \beta_{2} - 2 \beta_1 - 12) q^{60} + ( - 4 \beta_{2} - 8 \beta_1 - 53) q^{61} + ( - 20 \beta_{3} - 5 \beta_{2} + 10) q^{62} - 8 q^{64} + 6 \beta_{2} q^{65} + (8 \beta_{3} + 23 \beta_{2} - 7 \beta_1 - 60) q^{66} + (13 \beta_{2} + 26 \beta_1 - 39) q^{67} + ( - 4 \beta_{3} + 4 \beta_{2} + 2) q^{68} + ( - 19 \beta_{3} + 32 \beta_{2} + 2 \beta_1 + 30) q^{69} + (8 \beta_{3} - 68 \beta_{2} - 4) q^{71} + (8 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 24) q^{72} + (4 \beta_{2} + 8 \beta_1 - 33) q^{73} - \beta_{2} q^{74} + (14 \beta_{3} + 26 \beta_{2} + 2 \beta_1 - 48) q^{75} + (6 \beta_{2} + 12 \beta_1 + 2) q^{76} + (12 \beta_{3} + 12 \beta_{2} - 12 \beta_1 + 24) q^{78} + (17 \beta_{2} + 34 \beta_1 + 13) q^{79} + ( - 8 \beta_{3} + 8 \beta_{2} + 4) q^{80} + (35 \beta_{3} - 4 \beta_{2} + 20 \beta_1 - 42) q^{81} + (6 \beta_{2} + 12 \beta_1 - 12) q^{82} + ( - 16 \beta_{3} - 44 \beta_{2} + 8) q^{83} + (4 \beta_{2} + 8 \beta_1 + 19) q^{85} + ( - 24 \beta_{3} + 26 \beta_{2} + 12) q^{86} + (10 \beta_{3} + 40 \beta_{2} - 20 \beta_1 + 60) q^{87} + (10 \beta_{2} + 20 \beta_1 + 4) q^{88} + (66 \beta_{3} - 36 \beta_{2} - 33) q^{89} + ( - 16 \beta_{3} + 8 \beta_{2} - 13 \beta_1 - 24) q^{90} + (20 \beta_{3} - 14 \beta_{2} - 10) q^{92} + ( - 15 \beta_{3} - 30 \beta_{2} + 60) q^{93} + ( - 3 \beta_{2} - 6 \beta_1 + 90) q^{94} + (26 \beta_{3} - 35 \beta_{2} - 13) q^{95} + ( - 8 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{96} + (10 \beta_{2} + 20 \beta_1 + 72) q^{97} + ( - 29 \beta_{3} + 82 \beta_{2} + 49 \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 8 q^{4} - 4 q^{6} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 8 q^{4} - 4 q^{6} + 10 q^{9} - 16 q^{10} - 4 q^{12} - 32 q^{13} + 14 q^{15} + 16 q^{16} + 40 q^{18} - 4 q^{19} - 8 q^{22} + 8 q^{24} + 24 q^{25} + 80 q^{27} - 52 q^{30} - 20 q^{31} + 106 q^{33} + 16 q^{34} - 20 q^{36} - 4 q^{37} + 72 q^{39} + 32 q^{40} + 104 q^{43} + 58 q^{45} - 56 q^{46} + 8 q^{48} - 14 q^{51} + 64 q^{52} + 32 q^{54} - 236 q^{55} - 134 q^{57} + 80 q^{58} - 28 q^{60} - 212 q^{61} - 32 q^{64} - 224 q^{66} - 156 q^{67} + 82 q^{69} - 80 q^{72} - 132 q^{73} - 164 q^{75} + 8 q^{76} + 120 q^{78} + 52 q^{79} - 98 q^{81} - 48 q^{82} + 76 q^{85} + 260 q^{87} + 16 q^{88} - 128 q^{90} + 210 q^{93} + 360 q^{94} - 16 q^{96} + 288 q^{97} - 70 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} + 11x^{2} - 10x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 8\nu + 3 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{3} + 3\nu^{2} - 19\nu + 9 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} - 3\nu^{2} + 22\nu - 9 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 2\beta_{2} + 2\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -8\beta_{3} - 8\beta_{2} + 3\beta _1 - 3 \) 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
0.500000 3.07253i
0.500000 + 0.244099i
0.500000 + 3.07253i
0.500000 0.244099i
1.41421i −1.84521 2.36542i −2.00000 0.488198i −3.34521 + 2.60952i 0 2.82843i −2.19042 + 8.72938i 0.690416
197.2 1.41421i 2.84521 + 0.951206i −2.00000 6.14505i 1.34521 4.02373i 0 2.82843i 7.19042 + 5.41276i −8.69042
197.3 1.41421i −1.84521 + 2.36542i −2.00000 0.488198i −3.34521 2.60952i 0 2.82843i −2.19042 8.72938i 0.690416
197.4 1.41421i 2.84521 0.951206i −2.00000 6.14505i 1.34521 + 4.02373i 0 2.82843i 7.19042 5.41276i −8.69042
\(n\): 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.i 4
3.b odd 2 1 inner 294.3.b.i 4
7.b odd 2 1 294.3.b.e 4
7.c even 3 2 42.3.h.b 8
7.d odd 6 2 294.3.h.h 8
21.c even 2 1 294.3.b.e 4
21.g even 6 2 294.3.h.h 8
21.h odd 6 2 42.3.h.b 8
28.g odd 6 2 336.3.bn.g 8
84.n even 6 2 336.3.bn.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.h.b 8 7.c even 3 2
42.3.h.b 8 21.h odd 6 2
294.3.b.e 4 7.b odd 2 1
294.3.b.e 4 21.c even 2 1
294.3.b.i 4 1.a even 1 1 trivial
294.3.b.i 4 3.b odd 2 1 inner
294.3.h.h 8 7.d odd 6 2
294.3.h.h 8 21.g even 6 2
336.3.bn.g 8 28.g odd 6 2
336.3.bn.g 8 84.n even 6 2

Hecke kernels

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

\( T_{5}^{4} + 38T_{5}^{2} + 9 \) Copy content Toggle raw display
\( T_{13}^{2} + 16T_{13} - 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} - 3 T^{2} - 18 T + 81 \) Copy content Toggle raw display
$5$ \( T^{4} + 38T^{2} + 9 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 554 T^{2} + 74529 \) Copy content Toggle raw display
$13$ \( (T^{2} + 16 T - 24)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 38T^{2} + 9 \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T - 197)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 746 T^{2} + 31329 \) Copy content Toggle raw display
$29$ \( T^{4} + 2600 T^{2} + 810000 \) Copy content Toggle raw display
$31$ \( (T^{2} + 10 T - 525)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 936 T^{2} + 104976 \) Copy content Toggle raw display
$43$ \( (T^{2} - 52 T - 116)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 8298 T^{2} + \cdots + 15610401 \) Copy content Toggle raw display
$53$ \( T^{4} + 5382 T^{2} + \cdots + 6215049 \) Copy content Toggle raw display
$59$ \( T^{4} + 13178 T^{2} + \cdots + 30371121 \) Copy content Toggle raw display
$61$ \( (T^{2} + 106 T + 2457)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 78 T - 2197)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 18848 T^{2} + \cdots + 82301184 \) Copy content Toggle raw display
$73$ \( (T^{2} + 66 T + 737)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 26 T - 6189)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 9152 T^{2} + \cdots + 10036224 \) Copy content Toggle raw display
$89$ \( T^{4} + 29142 T^{2} + \cdots + 88115769 \) Copy content Toggle raw display
$97$ \( (T^{2} - 144 T + 2984)^{2} \) Copy content Toggle raw display
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