Properties

Label 49.22.a.d
Level $49$
Weight $22$
Character orbit 49.a
Self dual yes
Analytic conductor $136.944$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [49,22,Mod(1,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

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: yes
Analytic conductor: \(136.943898701\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3 x^{5} - 8712378 x^{4} - 2258837752 x^{3} + 15137375450784 x^{2} + \cdots - 48\!\cdots\!56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4}\cdot 5\cdot 7^{3} \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 428) q^{2} + ( - \beta_{2} + 15 \beta_1 - 43924) q^{3} + (\beta_{3} + \beta_{2} - 463 \beta_1 + 989963) q^{4} + (\beta_{5} + 2 \beta_{4} - \beta_{3} - 11 \beta_{2} - 2429 \beta_1 + 512069) q^{5} + ( - 2 \beta_{5} - 3 \beta_{4} - 51 \beta_{3} - 831 \beta_{2} + \cdots - 61686031) q^{6}+ \cdots + (147 \beta_{5} - 1194 \beta_{4} - 675 \beta_{3} + \cdots + 2437956270) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 428) q^{2} + ( - \beta_{2} + 15 \beta_1 - 43924) q^{3} + (\beta_{3} + \beta_{2} - 463 \beta_1 + 989963) q^{4} + (\beta_{5} + 2 \beta_{4} - \beta_{3} - 11 \beta_{2} - 2429 \beta_1 + 512069) q^{5} + ( - 2 \beta_{5} - 3 \beta_{4} - 51 \beta_{3} - 831 \beta_{2} + \cdots - 61686031) q^{6}+ \cdots + (11503989767739 \beta_{5} + \cdots + 57\!\cdots\!81) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2565 q^{2} - 263496 q^{3} + 5938389 q^{4} + 3065148 q^{5} - 369970578 q^{6} + 5228851671 q^{8} + 14619392190 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2565 q^{2} - 263496 q^{3} + 5938389 q^{4} + 3065148 q^{5} - 369970578 q^{6} + 5228851671 q^{8} + 14619392190 q^{9} + 43679516676 q^{10} + 113023233144 q^{11} - 434561122254 q^{12} - 182622640788 q^{13} - 127997950176 q^{15} + 8767702771665 q^{16} - 2935315661868 q^{17} + 52874653929165 q^{18} - 42449294530536 q^{19} - 7210207592280 q^{20} + 443428823757480 q^{22} + 601368739057152 q^{23} - 19\!\cdots\!26 q^{24}+ \cdots + 34\!\cdots\!84 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3 x^{5} - 8712378 x^{4} - 2258837752 x^{3} + 15137375450784 x^{2} + \cdots - 48\!\cdots\!56 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2001 \nu^{5} + 900929 \nu^{4} - 18334340390 \nu^{3} - 9273183570288 \nu^{2} + \cdots + 28\!\cdots\!44 ) / 88022801524224 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2001 \nu^{5} - 900929 \nu^{4} + 18334340390 \nu^{3} + 97295985094512 \nu^{2} + \cdots - 25\!\cdots\!88 ) / 88022801524224 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 51809 \nu^{5} - 137882197 \nu^{4} + 419145032050 \nu^{3} + \cdots - 90\!\cdots\!56 ) / 66017101143168 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 87753 \nu^{5} + 70954612 \nu^{4} + 595647307351 \nu^{3} - 181130479259850 \nu^{2} + \cdots + 13\!\cdots\!88 ) / 11002850190528 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 393\beta _1 + 2903931 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -42\beta_{5} - 243\beta_{4} + 1098\beta_{3} - 22026\beta_{2} + 5244270\beta _1 + 1139853608 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 20370\beta_{5} - 458433\beta_{4} + 7392624\beta_{3} - 1286892\beta_{2} + 5386802468\beta _1 + 15212131827534 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 394000110 \beta_{5} - 2020104513 \beta_{4} + 11366346812 \beta_{3} - 152612092960 \beta_{2} + 33108458957760 \beta _1 + 15\!\cdots\!18 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2752.23
1155.83
630.361
−649.563
−1673.85
−2212.01
−2324.23 31096.8 3.30489e6 −1.74638e7 −7.22761e7 0 −2.80705e9 −9.49334e9 4.05898e10
1.2 −727.834 −38526.8 −1.56741e6 3.45713e7 2.80411e7 0 2.66719e9 −8.97604e9 −2.51622e10
1.3 −202.361 −182183. −2.05620e6 −2.50577e7 3.68668e7 0 8.40477e8 2.27304e10 5.07070e9
1.4 1077.56 113831. −936010. 1.71654e7 1.22660e8 0 −3.26842e9 2.49708e9 1.84968e10
1.5 2101.85 −19144.5 2.32062e6 −3.88744e7 −4.02388e7 0 4.69702e8 −1.00938e10 −8.17081e10
1.6 2640.01 −168569. 4.87250e6 3.27243e7 −4.45023e8 0 7.32695e9 1.79551e10 8.63924e10
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.22.a.d 6
7.b odd 2 1 7.22.a.b 6
21.c even 2 1 63.22.a.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.22.a.b 6 7.b odd 2 1
49.22.a.d 6 1.a even 1 1 trivial
63.22.a.e 6 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2}^{6} - 2565 T_{2}^{5} - 5971038 T_{2}^{4} + 15611869368 T_{2}^{3} + 3162211853184 T_{2}^{2} + \cdots - 20\!\cdots\!60 \) Copy content Toggle raw display
\( T_{3}^{6} + 263496 T_{3}^{5} - 3975684696 T_{3}^{4} + \cdots + 80\!\cdots\!84 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 2565 T^{5} + \cdots - 20\!\cdots\!60 \) Copy content Toggle raw display
$3$ \( T^{6} + 263496 T^{5} + \cdots + 80\!\cdots\!84 \) Copy content Toggle raw display
$5$ \( T^{6} - 3065148 T^{5} + \cdots - 33\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 113023233144 T^{5} + \cdots - 20\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( T^{6} + 182622640788 T^{5} + \cdots + 95\!\cdots\!60 \) Copy content Toggle raw display
$17$ \( T^{6} + 2935315661868 T^{5} + \cdots - 31\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{6} + 42449294530536 T^{5} + \cdots - 78\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{6} - 601368739057152 T^{5} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 37\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots - 80\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 45\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 10\!\cdots\!80 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots - 85\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots - 19\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 83\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots - 18\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 10\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 44\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 46\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 11\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 10\!\cdots\!12 \) Copy content Toggle raw display
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