Properties

Label 315.10.a.m
Level $315$
Weight $10$
Character orbit 315.a
Self dual yes
Analytic conductor $162.236$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,10,Mod(1,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 315.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: \(162.236288392\)
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: \( x^{6} - 2x^{5} - 2794x^{4} - 2896x^{3} + 1850461x^{2} + 7006450x - 230581716 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 105)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 3) q^{2} + (\beta_{2} - 2 \beta_1 + 428) q^{4} + 625 q^{5} + 2401 q^{7} + (\beta_{4} - \beta_{3} + 7 \beta_{2} + \cdots + 1583) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 3) q^{2} + (\beta_{2} - 2 \beta_1 + 428) q^{4} + 625 q^{5} + 2401 q^{7} + (\beta_{4} - \beta_{3} + 7 \beta_{2} + \cdots + 1583) q^{8}+ \cdots + ( - 5764801 \beta_1 + 17294403) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 16 q^{2} + 2562 q^{4} + 3750 q^{5} + 14406 q^{7} + 8592 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 16 q^{2} + 2562 q^{4} + 3750 q^{5} + 14406 q^{7} + 8592 q^{8} + 10000 q^{10} - 17060 q^{11} + 78368 q^{13} + 38416 q^{14} + 1194674 q^{16} + 212740 q^{17} - 806620 q^{19} + 1601250 q^{20} - 826564 q^{22} + 542512 q^{23} + 2343750 q^{25} + 3016612 q^{26} + 6151362 q^{28} - 13648188 q^{29} + 9397812 q^{31} + 19680560 q^{32} - 15490336 q^{34} + 9003750 q^{35} + 1253060 q^{37} + 57043644 q^{38} + 5370000 q^{40} + 49035836 q^{41} + 46413816 q^{43} - 25242756 q^{44} + 143980160 q^{46} - 26203088 q^{47} + 34588806 q^{49} + 6250000 q^{50} + 349691552 q^{52} + 88807800 q^{53} - 10662500 q^{55} + 20629392 q^{56} + 162532192 q^{58} - 153792368 q^{59} + 183195388 q^{61} - 665443140 q^{62} + 1165541002 q^{64} + 48980000 q^{65} + 251552816 q^{67} - 476339804 q^{68} + 24010000 q^{70} - 150941500 q^{71} + 63087568 q^{73} - 1384051816 q^{74} + 1380142276 q^{76} - 40961060 q^{77} + 171607064 q^{79} + 746671250 q^{80} + 935307168 q^{82} + 1526136688 q^{83} + 132962500 q^{85} - 493447416 q^{86} - 578244268 q^{88} - 104312260 q^{89} + 188161568 q^{91} - 1867731944 q^{92} + 1548859320 q^{94} - 504137500 q^{95} + 293237016 q^{97} + 92236816 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 2794x^{4} - 2896x^{3} + 1850461x^{2} + 7006450x - 230581716 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4\nu - 931 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 65\nu^{4} - 831\nu^{3} - 147077\nu^{2} - 1440050\nu + 40535476 ) / 2392 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 65\nu^{4} - 3223\nu^{3} - 142293\nu^{2} + 2078582\nu + 45053964 ) / 2392 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -7\nu^{5} + 143\nu^{4} + 16581\nu^{3} - 278287\nu^{2} - 7000324\nu + 70037438 ) / 598 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4\beta _1 + 931 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + \beta_{3} + 2\beta_{2} + 1479\beta _1 + 3751 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 18\beta_{4} + 10\beta_{3} + 2151\beta_{2} + 10689\beta _1 + 1376964 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -65\beta_{5} - 2001\beta_{4} + 2573\beta_{3} + 8924\beta_{2} + 2562622\beta _1 + 10007632 \) 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
44.7693
29.7013
10.1694
−17.5906
−23.1181
−41.9314
−41.7693 0 1232.68 625.000 0 2401.00 −30102.2 0 −26105.8
1.2 −26.7013 0 200.962 625.000 0 2401.00 8305.13 0 −16688.3
1.3 −7.16942 0 −460.599 625.000 0 2401.00 6972.98 0 −4480.89
1.4 20.5906 0 −88.0274 625.000 0 2401.00 −12354.9 0 12869.1
1.5 26.1181 0 170.153 625.000 0 2401.00 −8928.38 0 16323.8
1.6 44.9314 0 1506.83 625.000 0 2401.00 44699.4 0 28082.2
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( -1 \)
\(7\) \( -1 \)

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 315.10.a.m 6
3.b odd 2 1 105.10.a.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.10.a.g 6 3.b odd 2 1
315.10.a.m 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 16T_{2}^{5} - 2689T_{2}^{4} + 36064T_{2}^{3} + 1674196T_{2}^{2} - 17729920T_{2} - 193212480 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(315))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 16 T^{5} + \cdots - 193212480 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T - 625)^{6} \) Copy content Toggle raw display
$7$ \( (T - 2401)^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots - 25\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots - 21\!\cdots\!20 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 14\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots - 34\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots - 28\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 14\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 73\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 51\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 26\!\cdots\!80 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 36\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 28\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 75\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 10\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 79\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots - 14\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots - 19\!\cdots\!44 \) Copy content Toggle raw display
show more
show less