Properties

Label 315.4.a.o
Level $315$
Weight $4$
Character orbit 315.a
Self dual yes
Analytic conductor $18.586$
Analytic rank $0$
Dimension $3$
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] = [315,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(18.5856016518\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.22952.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 18x + 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - 2 \beta_1 + 5) q^{4} + 5 q^{5} + 7 q^{7} + (2 \beta_{2} - 5 \beta_1 + 23) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - 2 \beta_1 + 5) q^{4} + 5 q^{5} + 7 q^{7} + (2 \beta_{2} - 5 \beta_1 + 23) q^{8} + ( - 5 \beta_1 + 5) q^{10} + (4 \beta_{2} + 6 \beta_1 - 10) q^{11} + ( - 4 \beta_{2} - 2 \beta_1 + 2) q^{13} + ( - 7 \beta_1 + 7) q^{14} + ( - 3 \beta_{2} - 24 \beta_1 + 47) q^{16} + ( - 12 \beta_1 + 82) q^{17} + (8 \beta_{2} - 6 \beta_1 - 28) q^{19} + (5 \beta_{2} - 10 \beta_1 + 25) q^{20} + ( - 6 \beta_{2} - 8 \beta_1 - 74) q^{22} + ( - 12 \beta_{2} + 12 \beta_1 + 10) q^{23} + 25 q^{25} + (2 \beta_{2} + 20 \beta_1 + 18) q^{26} + (7 \beta_{2} - 14 \beta_1 + 35) q^{28} + ( - 16 \beta_{2} + 32 \beta_1 - 16) q^{29} + (10 \beta_1 - 124) q^{31} + (8 \beta_{2} - 13 \beta_1 + 145) q^{32} + (12 \beta_{2} - 94 \beta_1 + 226) q^{34} + 35 q^{35} + ( - 40 \beta_1 + 114) q^{37} + (6 \beta_{2} - 26 \beta_1 + 60) q^{38} + (10 \beta_{2} - 25 \beta_1 + 115) q^{40} + ( - 20 \beta_{2} + 52 \beta_1 + 198) q^{41} + ( - 36 \beta_{2} + 52 \beta_1 - 28) q^{43} + ( - 24 \beta_{2} + 54 \beta_1 + 90) q^{44} + ( - 12 \beta_{2} + 74 \beta_1 - 158) q^{46} + (4 \beta_{2} + 8 \beta_1 + 308) q^{47} + 49 q^{49} + ( - 25 \beta_1 + 25) q^{50} + (12 \beta_{2} + 6 \beta_1 - 234) q^{52} + ( - 20 \beta_{2} - 54 \beta_1 + 124) q^{53} + (20 \beta_{2} + 30 \beta_1 - 50) q^{55} + (14 \beta_{2} - 35 \beta_1 + 161) q^{56} + ( - 32 \beta_{2} + 144 \beta_1 - 432) q^{58} + (72 \beta_{2} - 16 \beta_1 + 124) q^{59} + (120 \beta_1 + 310) q^{61} + ( - 10 \beta_{2} + 134 \beta_1 - 244) q^{62} + (37 \beta_{2} - 14 \beta_1 - 59) q^{64} + ( - 20 \beta_{2} - 10 \beta_1 + 10) q^{65} + (16 \beta_{2} + 208 \beta_1 - 92) q^{67} + (94 \beta_{2} - 296 \beta_1 + 722) q^{68} + ( - 35 \beta_1 + 35) q^{70} + ( - 4 \beta_{2} + 66 \beta_1 + 118) q^{71} + ( - 44 \beta_{2} - 22 \beta_1 - 82) q^{73} + (40 \beta_{2} - 154 \beta_1 + 594) q^{74} + ( - 38 \beta_{2} - 74 \beta_1 + 608) q^{76} + (28 \beta_{2} + 42 \beta_1 - 70) q^{77} + (108 \beta_{2} + 124 \beta_1 - 240) q^{79} + ( - 15 \beta_{2} - 120 \beta_1 + 235) q^{80} + ( - 52 \beta_{2} - 26 \beta_1 - 466) q^{82} + ( - 76 \beta_{2} - 128 \beta_1 + 384) q^{83} + ( - 60 \beta_1 + 410) q^{85} + ( - 52 \beta_{2} + 296 \beta_1 - 724) q^{86} + ( - 6 \beta_{2} + 172 \beta_1 - 14) q^{88} + ( - 32 \beta_{2} + 88 \beta_1 - 66) q^{89} + ( - 28 \beta_{2} - 14 \beta_1 + 14) q^{91} + (22 \beta_{2} + 208 \beta_1 - 1150) q^{92} + ( - 8 \beta_{2} - 324 \beta_1 + 220) q^{94} + (40 \beta_{2} - 30 \beta_1 - 140) q^{95} + (12 \beta_{2} + 206 \beta_1 - 1090) q^{97} + ( - 49 \beta_1 + 49) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 14 q^{4} + 15 q^{5} + 21 q^{7} + 66 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} + 14 q^{4} + 15 q^{5} + 21 q^{7} + 66 q^{8} + 10 q^{10} - 20 q^{11} + 14 q^{14} + 114 q^{16} + 234 q^{17} - 82 q^{19} + 70 q^{20} - 236 q^{22} + 30 q^{23} + 75 q^{25} + 76 q^{26} + 98 q^{28} - 32 q^{29} - 362 q^{31} + 430 q^{32} + 596 q^{34} + 105 q^{35} + 302 q^{37} + 160 q^{38} + 330 q^{40} + 626 q^{41} - 68 q^{43} + 300 q^{44} - 412 q^{46} + 936 q^{47} + 147 q^{49} + 50 q^{50} - 684 q^{52} + 298 q^{53} - 100 q^{55} + 462 q^{56} - 1184 q^{58} + 428 q^{59} + 1050 q^{61} - 608 q^{62} - 154 q^{64} - 52 q^{67} + 1964 q^{68} + 70 q^{70} + 416 q^{71} - 312 q^{73} + 1668 q^{74} + 1712 q^{76} - 140 q^{77} - 488 q^{79} + 570 q^{80} - 1476 q^{82} + 948 q^{83} + 1170 q^{85} - 1928 q^{86} + 124 q^{88} - 142 q^{89} - 3220 q^{92} + 328 q^{94} - 410 q^{95} - 3052 q^{97} + 98 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 18x + 14 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 12 \) 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
4.37989
0.770205
−4.15010
−3.37989 0 3.42368 5.00000 0 7.00000 15.4675 0 −16.8995
1.2 0.229795 0 −7.94719 5.00000 0 7.00000 −3.66459 0 1.14898
1.3 5.15010 0 18.5235 5.00000 0 7.00000 54.1971 0 25.7505
\(n\): e.g. 2-40 or 990-1000
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 315.4.a.o yes 3
3.b odd 2 1 315.4.a.n 3
5.b even 2 1 1575.4.a.bb 3
7.b odd 2 1 2205.4.a.bl 3
15.d odd 2 1 1575.4.a.be 3
21.c even 2 1 2205.4.a.bk 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.4.a.n 3 3.b odd 2 1
315.4.a.o yes 3 1.a even 1 1 trivial
1575.4.a.bb 3 5.b even 2 1
1575.4.a.be 3 15.d odd 2 1
2205.4.a.bk 3 21.c even 2 1
2205.4.a.bl 3 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 2T_{2}^{2} - 17T_{2} + 4 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(315))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 2 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T - 5)^{3} \) Copy content Toggle raw display
$7$ \( (T - 7)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 20 T^{2} + \cdots - 32160 \) Copy content Toggle raw display
$13$ \( T^{3} - 1748T - 17328 \) Copy content Toggle raw display
$17$ \( T^{3} - 234 T^{2} + \cdots - 282328 \) Copy content Toggle raw display
$19$ \( T^{3} + 82 T^{2} + \cdots + 15280 \) Copy content Toggle raw display
$23$ \( T^{3} - 30 T^{2} + \cdots - 378280 \) Copy content Toggle raw display
$29$ \( T^{3} + 32 T^{2} + \cdots + 409600 \) Copy content Toggle raw display
$31$ \( T^{3} + 362 T^{2} + \cdots + 1543664 \) Copy content Toggle raw display
$37$ \( T^{3} - 302 T^{2} + \cdots + 1425496 \) Copy content Toggle raw display
$41$ \( T^{3} - 626 T^{2} + \cdots + 16079208 \) Copy content Toggle raw display
$43$ \( T^{3} + 68 T^{2} + \cdots - 10742464 \) Copy content Toggle raw display
$47$ \( T^{3} - 936 T^{2} + \cdots - 29520128 \) Copy content Toggle raw display
$53$ \( T^{3} - 298 T^{2} + \cdots + 19383056 \) Copy content Toggle raw display
$59$ \( T^{3} - 428 T^{2} + \cdots + 229584064 \) Copy content Toggle raw display
$61$ \( T^{3} - 1050 T^{2} + \cdots + 63221000 \) Copy content Toggle raw display
$67$ \( T^{3} + 52 T^{2} + \cdots - 93054016 \) Copy content Toggle raw display
$71$ \( T^{3} - 416 T^{2} + \cdots + 14345328 \) Copy content Toggle raw display
$73$ \( T^{3} + 312 T^{2} + \cdots - 43935536 \) Copy content Toggle raw display
$79$ \( T^{3} + 488 T^{2} + \cdots - 282894592 \) Copy content Toggle raw display
$83$ \( T^{3} - 948 T^{2} + \cdots + 431507200 \) Copy content Toggle raw display
$89$ \( T^{3} + 142 T^{2} + \cdots + 19655976 \) Copy content Toggle raw display
$97$ \( T^{3} + 3052 T^{2} + \cdots + 204161472 \) Copy content Toggle raw display
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