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Properties

Label 315.4.j.c
Level $315$
Weight $4$
Character orbit 315.j
Analytic conductor $18.586$
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] = [315,4,Mod(46,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.46");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 315.j (of order \(3\), degree \(2\), 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: \(18.5856016518\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} + 3 \beta_{2} + \beta_1) q^{2} + ( - 3 \beta_{2} - 6 \beta_1 - 3) q^{4} - 5 \beta_{2} q^{5} + ( - 2 \beta_{3} - 5 \beta_{2} + \cdots + 3) q^{7}+ \cdots + ( - 13 \beta_{3} - 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 3 \beta_{2} + \beta_1) q^{2} + ( - 3 \beta_{2} - 6 \beta_1 - 3) q^{4} - 5 \beta_{2} q^{5} + ( - 2 \beta_{3} - 5 \beta_{2} + \cdots + 3) q^{7}+ \cdots + (644 \beta_{3} - 301 \beta_{2} + \cdots - 917) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{2} - 6 q^{4} + 10 q^{5} + 22 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{2} - 6 q^{4} + 10 q^{5} + 22 q^{7} - 12 q^{8} + 30 q^{10} + 28 q^{11} - 72 q^{13} - 84 q^{14} + 14 q^{16} - 76 q^{17} - 160 q^{19} - 60 q^{20} - 88 q^{22} - 22 q^{23} - 50 q^{25} + 148 q^{26} + 138 q^{28} + 500 q^{29} + 132 q^{31} + 42 q^{32} + 888 q^{34} - 20 q^{35} + 416 q^{37} - 376 q^{38} - 30 q^{40} + 212 q^{41} - 1332 q^{43} - 156 q^{44} + 402 q^{46} - 196 q^{47} - 230 q^{49} + 300 q^{50} + 348 q^{52} - 952 q^{53} + 280 q^{55} + 714 q^{56} - 510 q^{58} + 840 q^{59} - 98 q^{61} + 8 q^{62} - 1204 q^{64} - 180 q^{65} + 1286 q^{67} - 1524 q^{68} + 210 q^{70} - 2128 q^{71} + 172 q^{73} + 1792 q^{74} - 288 q^{76} + 724 q^{77} + 1240 q^{79} - 70 q^{80} - 438 q^{82} + 3812 q^{83} - 760 q^{85} + 2018 q^{86} - 604 q^{88} + 650 q^{89} - 996 q^{91} - 5484 q^{92} - 332 q^{94} + 800 q^{95} - 1256 q^{97} - 3066 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
0.707107 − 1.22474i
−0.707107 + 1.22474i
0.707107 + 1.22474i
−0.707107 − 1.22474i
−2.20711 − 3.82282i 0 −5.74264 + 9.94655i 2.50000 + 4.33013i 0 16.1066 − 9.14207i 15.3848 0 11.0355 − 19.1141i
46.2 −0.792893 − 1.37333i 0 2.74264 − 4.75039i 2.50000 + 4.33013i 0 −5.10660 + 17.8023i −21.3848 0 3.96447 − 6.86666i
226.1 −2.20711 + 3.82282i 0 −5.74264 − 9.94655i 2.50000 − 4.33013i 0 16.1066 + 9.14207i 15.3848 0 11.0355 + 19.1141i
226.2 −0.792893 + 1.37333i 0 2.74264 + 4.75039i 2.50000 − 4.33013i 0 −5.10660 − 17.8023i −21.3848 0 3.96447 + 6.86666i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.4.j.c 4
3.b odd 2 1 35.4.e.b ✓ 4
7.c even 3 1 inner 315.4.j.c 4
7.c even 3 1 2205.4.a.bf 2
7.d odd 6 1 2205.4.a.bg 2
12.b even 2 1 560.4.q.i 4
15.d odd 2 1 175.4.e.c 4
15.e even 4 2 175.4.k.c 8
21.c even 2 1 245.4.e.l 4
21.g even 6 1 245.4.a.h 2
21.g even 6 1 245.4.e.l 4
21.h odd 6 1 35.4.e.b ✓ 4
21.h odd 6 1 245.4.a.g 2
84.n even 6 1 560.4.q.i 4
105.o odd 6 1 175.4.e.c 4
105.o odd 6 1 1225.4.a.x 2
105.p even 6 1 1225.4.a.v 2
105.x even 12 2 175.4.k.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.b ✓ 4 3.b odd 2 1
35.4.e.b ✓ 4 21.h odd 6 1
175.4.e.c 4 15.d odd 2 1
175.4.e.c 4 105.o odd 6 1
175.4.k.c 8 15.e even 4 2
175.4.k.c 8 105.x even 12 2
245.4.a.g 2 21.h odd 6 1
245.4.a.h 2 21.g even 6 1
245.4.e.l 4 21.c even 2 1
245.4.e.l 4 21.g even 6 1
315.4.j.c 4 1.a even 1 1 trivial
315.4.j.c 4 7.c even 3 1 inner
560.4.q.i 4 12.b even 2 1
560.4.q.i 4 84.n even 6 1
1225.4.a.v 2 105.p even 6 1
1225.4.a.x 2 105.o odd 6 1
2205.4.a.bf 2 7.c even 3 1
2205.4.a.bg 2 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(315, [\chi])\):

\( T_{2}^{4} + 6T_{2}^{3} + 29T_{2}^{2} + 42T_{2} + 49 \) Copy content Toggle raw display
\( T_{13}^{2} + 36T_{13} + 124 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 6 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 22 T^{3} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{4} - 28 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( (T^{2} + 36 T + 124)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 76 T^{3} + \cdots + 19254544 \) Copy content Toggle raw display
$19$ \( T^{4} + 160 T^{3} + \cdots + 25482304 \) Copy content Toggle raw display
$23$ \( T^{4} + 22 T^{3} + \cdots + 742944049 \) Copy content Toggle raw display
$29$ \( (T^{2} - 250 T + 8425)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 132 T^{3} + \cdots + 244734736 \) Copy content Toggle raw display
$37$ \( T^{4} - 416 T^{3} + \cdots + 39337984 \) Copy content Toggle raw display
$41$ \( (T^{2} - 106 T + 1009)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 666 T + 110839)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 196 T^{3} + \cdots + 1993744 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 34688317504 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 1217172544 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 419125465201 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 163157021329 \) Copy content Toggle raw display
$71$ \( (T^{2} + 1064 T + 38024)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 172 T^{3} + \cdots + 418284304 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 85736524864 \) Copy content Toggle raw display
$83$ \( (T^{2} - 1906 T + 908159)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 10884122929 \) Copy content Toggle raw display
$97$ \( (T^{2} + 628 T - 401404)^{2} \) Copy content Toggle raw display
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