Properties

Label 315.10.a.j
Level $315$
Weight $10$
Character orbit 315.a
Self dual yes
Analytic conductor $162.236$
Analytic rank $0$
Dimension $5$
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: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 1694x^{3} + 7420x^{2} + 583584x - 2721600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3}\cdot 3\cdot 5 \)
Twist minimal: no (minimal twist has level 35)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{3} - 4 \beta_1 + 168) q^{4} + 625 q^{5} + 2401 q^{7} + ( - 2 \beta_{4} + 6 \beta_{3} + \cdots + 2816) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{3} - 4 \beta_1 + 168) q^{4} + 625 q^{5} + 2401 q^{7} + ( - 2 \beta_{4} + 6 \beta_{3} + \cdots + 2816) q^{8}+ \cdots - 5764801 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} + 832 q^{4} + 3125 q^{5} + 12005 q^{7} + 14136 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{2} + 832 q^{4} + 3125 q^{5} + 12005 q^{7} + 14136 q^{8} - 1250 q^{10} - 10312 q^{11} + 158638 q^{13} - 4802 q^{14} - 526696 q^{16} - 31614 q^{17} + 1655376 q^{19} + 520000 q^{20} + 3659464 q^{22} - 796104 q^{23} + 1953125 q^{25} - 10219060 q^{26} + 1997632 q^{28} + 3353726 q^{29} + 2678120 q^{31} + 130400 q^{32} + 21548612 q^{34} + 7503125 q^{35} + 50994846 q^{37} + 34907600 q^{38} + 8835000 q^{40} - 6330194 q^{41} + 6149468 q^{43} + 71225892 q^{44} - 142845472 q^{46} + 6897780 q^{47} + 28824005 q^{49} - 781250 q^{50} - 22199468 q^{52} + 70886738 q^{53} - 6445000 q^{55} + 33940536 q^{56} - 31955924 q^{58} + 152335168 q^{59} + 257015698 q^{61} + 215582768 q^{62} - 193988432 q^{64} + 99148750 q^{65} + 133467828 q^{67} + 268709380 q^{68} - 3001250 q^{70} - 522788960 q^{71} + 159370858 q^{73} + 244178868 q^{74} - 345056616 q^{76} - 24759112 q^{77} + 464174900 q^{79} - 329185000 q^{80} + 308947620 q^{82} - 2207636832 q^{83} - 19758750 q^{85} - 694560616 q^{86} + 930528736 q^{88} + 1106708326 q^{89} + 380889838 q^{91} - 205160216 q^{92} - 3602501744 q^{94} + 1034610000 q^{95} + 1956142254 q^{97} - 11529602 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 1694x^{3} + 7420x^{2} + 583584x - 2721600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 52\nu^{3} - 1022\nu^{2} - 47768\nu + 163608 ) / 712 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 4\nu - 680 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -7\nu^{4} - 8\nu^{3} + 9290\nu^{2} - 12368\nu - 1595240 ) / 712 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 4\beta _1 + 680 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{4} - 6\beta_{3} + 14\beta_{2} + 998\beta _1 - 2816 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -104\beta_{4} + 1334\beta_{3} - 16\beta_{2} - 8216\beta _1 + 677784 \) 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
32.5415
22.7661
4.68049
−21.5262
−36.4619
−32.5415 0 546.947 625.000 0 2401.00 −1137.23 0 −20338.4
1.2 −22.7661 0 6.29742 625.000 0 2401.00 11512.9 0 −14228.8
1.3 −4.68049 0 −490.093 625.000 0 2401.00 4690.29 0 −2925.31
1.4 21.5262 0 −48.6242 625.000 0 2401.00 −12068.1 0 13453.9
1.5 36.4619 0 817.473 625.000 0 2401.00 11138.1 0 22788.7
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.j 5
3.b odd 2 1 35.10.a.d 5
15.d odd 2 1 175.10.a.f 5
15.e even 4 2 175.10.b.f 10
21.c even 2 1 245.10.a.f 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.10.a.d 5 3.b odd 2 1
175.10.a.f 5 15.d odd 2 1
175.10.b.f 10 15.e even 4 2
245.10.a.f 5 21.c even 2 1
315.10.a.j 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} + 2T_{2}^{4} - 1694T_{2}^{3} - 7420T_{2}^{2} + 583584T_{2} + 2721600 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(315))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + 2 T^{4} + \cdots + 2721600 \) Copy content Toggle raw display
$3$ \( T^{5} \) Copy content Toggle raw display
$5$ \( (T - 625)^{5} \) Copy content Toggle raw display
$7$ \( (T - 2401)^{5} \) Copy content Toggle raw display
$11$ \( T^{5} + \cdots + 27\!\cdots\!08 \) Copy content Toggle raw display
$13$ \( T^{5} + \cdots - 43\!\cdots\!10 \) Copy content Toggle raw display
$17$ \( T^{5} + \cdots - 77\!\cdots\!66 \) Copy content Toggle raw display
$19$ \( T^{5} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{5} + \cdots - 87\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{5} + \cdots + 10\!\cdots\!50 \) Copy content Toggle raw display
$31$ \( T^{5} + \cdots + 97\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots + 46\!\cdots\!92 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots + 20\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( T^{5} + \cdots + 72\!\cdots\!80 \) Copy content Toggle raw display
$47$ \( T^{5} + \cdots - 84\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{5} + \cdots + 37\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots - 43\!\cdots\!12 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots - 32\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots - 60\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots + 42\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots + 11\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots - 53\!\cdots\!94 \) Copy content Toggle raw display
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