Properties

Label 315.10.a.o
Level $315$
Weight $10$
Character orbit 315.a
Self dual yes
Analytic conductor $162.236$
Analytic rank $0$
Dimension $8$
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] = [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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 2943 x^{6} + 5227 x^{5} + 2674143 x^{4} - 7013491 x^{3} - 759677093 x^{2} + \cdots + 51239796804 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 3) q^{2} + (\beta_{2} + 5 \beta_1 + 233) q^{4} - 625 q^{5} + 2401 q^{7} + (\beta_{5} + \beta_{4} + 9 \beta_{2} + \cdots + 2582) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 3) q^{2} + (\beta_{2} + 5 \beta_1 + 233) q^{4} - 625 q^{5} + 2401 q^{7} + (\beta_{5} + \beta_{4} + 9 \beta_{2} + \cdots + 2582) q^{8}+ \cdots + (5764801 \beta_1 + 17294403) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 25 q^{2} + 1869 q^{4} - 5000 q^{5} + 19208 q^{7} + 20775 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 25 q^{2} + 1869 q^{4} - 5000 q^{5} + 19208 q^{7} + 20775 q^{8} - 15625 q^{10} - 7600 q^{11} + 82954 q^{13} + 60025 q^{14} - 211855 q^{16} + 183750 q^{17} - 358708 q^{19} - 1168125 q^{20} - 256480 q^{22} + 1628250 q^{23} + 3125000 q^{25} + 5736650 q^{26} + 4487469 q^{28} + 9632450 q^{29} - 4176606 q^{31} + 12988175 q^{32} - 6755408 q^{34} - 12005000 q^{35} - 26383078 q^{37} + 20949050 q^{38} - 12984375 q^{40} + 28619500 q^{41} - 32860758 q^{43} + 23594400 q^{44} - 18837272 q^{46} + 98763150 q^{47} + 46118408 q^{49} + 9765625 q^{50} - 121422998 q^{52} + 160178600 q^{53} + 4750000 q^{55} + 49880775 q^{56} - 169757458 q^{58} + 173756800 q^{59} - 150415226 q^{61} - 18430000 q^{62} - 234286047 q^{64} - 51846250 q^{65} - 161003638 q^{67} + 82201600 q^{68} - 37515625 q^{70} + 203106250 q^{71} - 86008694 q^{73} + 18467100 q^{74} + 878032874 q^{76} - 18247600 q^{77} + 189530040 q^{79} + 132409375 q^{80} + 658559094 q^{82} - 24719700 q^{83} - 114843750 q^{85} + 410747450 q^{86} + 1560789440 q^{88} + 1311421900 q^{89} + 199172554 q^{91} - 233402000 q^{92} + 2107673474 q^{94} + 224192500 q^{95} + 2176527554 q^{97} + 144120025 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 2943 x^{6} + 5227 x^{5} + 2674143 x^{4} - 7013491 x^{3} - 759677093 x^{2} + \cdots + 51239796804 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 736 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 67 \nu^{7} - 59216 \nu^{6} + 831075 \nu^{5} + 108855364 \nu^{4} - 1392176191 \nu^{3} + \cdots + 3565684744244 ) / 187504384 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1179 \nu^{7} - 7440 \nu^{6} + 2341909 \nu^{5} + 36125308 \nu^{4} - 907999177 \nu^{3} + \cdots + 5986461701612 ) / 375008768 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1179 \nu^{7} + 7440 \nu^{6} - 2341909 \nu^{5} - 36125308 \nu^{4} + 1283007945 \nu^{3} + \cdots - 5612577959916 ) / 375008768 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5569 \nu^{7} - 24496 \nu^{6} - 14004175 \nu^{5} + 41835468 \nu^{4} + 9939069163 \nu^{3} + \cdots + 626401057148 ) / 375008768 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3307 \nu^{7} + 50688 \nu^{6} - 9004117 \nu^{5} - 152876380 \nu^{4} + 7143843033 \nu^{3} + \cdots - 19198705417580 ) / 750017536 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 736 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + 1125\beta _1 - 997 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -16\beta_{7} + 4\beta_{6} + 7\beta_{5} + 3\beta_{4} - 4\beta_{3} + 1509\beta_{2} - 1729\beta _1 + 827321 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 416 \beta_{7} + 8 \beta_{6} + 1998 \beta_{5} + 1446 \beta_{4} - 56 \beta_{3} + 1508 \beta_{2} + \cdots - 1667322 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 36480 \beta_{7} + 7568 \beta_{6} + 21110 \beta_{5} + 4390 \beta_{4} - 11488 \beta_{3} + \cdots + 1040182890 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1086368 \beta_{7} + 90696 \beta_{6} + 3279859 \beta_{5} + 1848267 \beta_{4} - 161304 \beta_{3} + \cdots - 1923352759 \) 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
−36.5031
−35.3875
−18.4124
−8.24526
11.9139
18.5924
31.0392
38.0027
−33.5031 0 610.455 −625.000 0 2401.00 −3298.54 0 20939.4
1.2 −32.3875 0 536.948 −625.000 0 2401.00 −808.016 0 20242.2
1.3 −15.4124 0 −274.458 −625.000 0 2401.00 12121.2 0 9632.74
1.4 −5.24526 0 −484.487 −625.000 0 2401.00 5226.83 0 3278.29
1.5 14.9139 0 −289.575 −625.000 0 2401.00 −11954.6 0 −9321.19
1.6 21.5924 0 −45.7679 −625.000 0 2401.00 −12043.6 0 −13495.3
1.7 34.0392 0 646.665 −625.000 0 2401.00 4583.89 0 −21274.5
1.8 41.0027 0 1169.22 −625.000 0 2401.00 26947.8 0 −25626.7
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.o yes 8
3.b odd 2 1 315.10.a.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.10.a.n 8 3.b odd 2 1
315.10.a.o yes 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 25 T_{2}^{7} - 2670 T_{2}^{6} + 56500 T_{2}^{5} + 2205048 T_{2}^{4} - 37060000 T_{2}^{3} + \cdots + 39425937408 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(315))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots + 39425937408 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T + 625)^{8} \) Copy content Toggle raw display
$7$ \( (T - 2401)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots - 16\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots - 24\!\cdots\!08 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots - 22\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots - 17\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots - 12\!\cdots\!48 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots - 34\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots - 31\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 11\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 29\!\cdots\!92 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots - 21\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 31\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 84\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots - 19\!\cdots\!68 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots - 44\!\cdots\!96 \) Copy content Toggle raw display
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