Properties

Label 315.4.a.p
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: 3.3.14360.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 17x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - \beta_1 + 4) q^{4} - 5 q^{5} + 7 q^{7} + (3 \beta_{2} - \beta_1 + 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - \beta_1 + 4) q^{4} - 5 q^{5} + 7 q^{7} + (3 \beta_{2} - \beta_1 + 4) q^{8} + (5 \beta_1 - 5) q^{10} + ( - \beta_{2} - 3 \beta_1 + 25) q^{11} + (5 \beta_{2} - 13 \beta_1 + 13) q^{13} + ( - 7 \beta_1 + 7) q^{14} + ( - \beta_{2} - 11 \beta_1 - 26) q^{16} + ( - 11 \beta_{2} - 13 \beta_1 + 21) q^{17} + (6 \beta_{2} - 10 \beta_1 + 54) q^{19} + ( - 5 \beta_{2} + 5 \beta_1 - 20) q^{20} + (\beta_{2} - 20 \beta_1 + 61) q^{22} + ( - 2 \beta_{2} + 14 \beta_1 + 42) q^{23} + 25 q^{25} + (23 \beta_{2} - 38 \beta_1 + 141) q^{26} + (7 \beta_{2} - 7 \beta_1 + 28) q^{28} + ( - 17 \beta_{2} - 19 \beta_1 - 105) q^{29} + ( - 4 \beta_{2} + 24 \beta_1 + 108) q^{31} + ( - 15 \beta_{2} + 39 \beta_1 + 66) q^{32} + ( - 9 \beta_{2} + 34 \beta_1 + 197) q^{34} - 35 q^{35} + ( - 12 \beta_{2} + 16 \beta_1 - 14) q^{37} + (22 \beta_{2} - 84 \beta_1 + 146) q^{38} + ( - 15 \beta_{2} + 5 \beta_1 - 20) q^{40} + ( - 2 \beta_{2} - 10 \beta_1 - 120) q^{41} + ( - 34 \beta_{2} + 30 \beta_1 + 6) q^{43} + (30 \beta_{2} - 42 \beta_1 + 78) q^{44} + ( - 18 \beta_{2} - 32 \beta_1 - 106) q^{46} + (13 \beta_{2} + 51 \beta_1 + 239) q^{47} + 49 q^{49} + ( - 25 \beta_1 + 25) q^{50} + (44 \beta_{2} - 152 \beta_1 + 386) q^{52} + (22 \beta_{2} + 130 \beta_1 - 44) q^{53} + (5 \beta_{2} + 15 \beta_1 - 125) q^{55} + (21 \beta_{2} - 7 \beta_1 + 28) q^{56} + ( - 15 \beta_{2} + 190 \beta_1 + 155) q^{58} + ( - 48 \beta_{2} + 176 \beta_1 + 76) q^{59} + ( - 26 \beta_{2} - 34 \beta_1 + 416) q^{61} + ( - 32 \beta_{2} - 88 \beta_1 - 144) q^{62} + ( - 61 \beta_{2} + 97 \beta_1 - 110) q^{64} + ( - 25 \beta_{2} + 65 \beta_1 - 65) q^{65} + (108 \beta_{2} - 12 \beta_1 + 32) q^{67} + (36 \beta_{2} - 48 \beta_1 - 318) q^{68} + (35 \beta_1 - 35) q^{70} + (40 \beta_{2} - 72 \beta_1 + 32) q^{71} + (76 \beta_{2} + 124 \beta_1 + 78) q^{73} + ( - 40 \beta_{2} + 74 \beta_1 - 154) q^{74} + (80 \beta_{2} - 176 \beta_1 + 572) q^{76} + ( - 7 \beta_{2} - 21 \beta_1 + 175) q^{77} + ( - 89 \beta_{2} - 83 \beta_1 - 315) q^{79} + (5 \beta_{2} + 55 \beta_1 + 130) q^{80} + (6 \beta_{2} + 130 \beta_1 - 4) q^{82} + ( - 8 \beta_{2} + 160 \beta_1 + 556) q^{83} + (55 \beta_{2} + 65 \beta_1 - 105) q^{85} + ( - 98 \beta_{2} + 164 \beta_1 - 222) q^{86} + (94 \beta_{2} - 68 \beta_1 - 38) q^{88} + (82 \beta_{2} + 98 \beta_1 - 108) q^{89} + (35 \beta_{2} - 91 \beta_1 + 91) q^{91} + (12 \beta_{2} + 84 \beta_1 - 36) q^{92} + ( - 25 \beta_{2} - 304 \beta_1 - 361) q^{94} + ( - 30 \beta_{2} + 50 \beta_1 - 270) q^{95} + ( - 65 \beta_{2} - 87 \beta_1 + 55) q^{97} + ( - 49 \beta_1 + 49) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 13 q^{4} - 15 q^{5} + 21 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 13 q^{4} - 15 q^{5} + 21 q^{7} + 15 q^{8} - 15 q^{10} + 74 q^{11} + 44 q^{13} + 21 q^{14} - 79 q^{16} + 52 q^{17} + 168 q^{19} - 65 q^{20} + 184 q^{22} + 124 q^{23} + 75 q^{25} + 446 q^{26} + 91 q^{28} - 332 q^{29} + 320 q^{31} + 183 q^{32} + 582 q^{34} - 105 q^{35} - 54 q^{37} + 460 q^{38} - 75 q^{40} - 362 q^{41} - 16 q^{43} + 264 q^{44} - 336 q^{46} + 730 q^{47} + 147 q^{49} + 75 q^{50} + 1202 q^{52} - 110 q^{53} - 370 q^{55} + 105 q^{56} + 450 q^{58} + 180 q^{59} + 1222 q^{61} - 464 q^{62} - 391 q^{64} - 220 q^{65} + 204 q^{67} - 918 q^{68} - 105 q^{70} + 136 q^{71} + 310 q^{73} - 502 q^{74} + 1796 q^{76} + 518 q^{77} - 1034 q^{79} + 395 q^{80} - 6 q^{82} + 1660 q^{83} - 260 q^{85} - 764 q^{86} - 20 q^{88} - 242 q^{89} + 308 q^{91} - 96 q^{92} - 1108 q^{94} - 840 q^{95} + 100 q^{97} + 147 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 17x - 14 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 11 \) 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
4.48565
−0.861086
−3.62456
−3.48565 0 4.14976 −5.00000 0 7.00000 13.4206 0 17.4283
1.2 1.86109 0 −4.53636 −5.00000 0 7.00000 −23.3312 0 −9.30543
1.3 4.62456 0 13.3866 −5.00000 0 7.00000 24.9107 0 −23.1228
\(n\): 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.p 3
3.b odd 2 1 35.4.a.c 3
5.b even 2 1 1575.4.a.ba 3
7.b odd 2 1 2205.4.a.bm 3
12.b even 2 1 560.4.a.u 3
15.d odd 2 1 175.4.a.f 3
15.e even 4 2 175.4.b.e 6
21.c even 2 1 245.4.a.l 3
21.g even 6 2 245.4.e.n 6
21.h odd 6 2 245.4.e.m 6
24.f even 2 1 2240.4.a.bv 3
24.h odd 2 1 2240.4.a.bt 3
105.g even 2 1 1225.4.a.y 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.c 3 3.b odd 2 1
175.4.a.f 3 15.d odd 2 1
175.4.b.e 6 15.e even 4 2
245.4.a.l 3 21.c even 2 1
245.4.e.m 6 21.h odd 6 2
245.4.e.n 6 21.g even 6 2
315.4.a.p 3 1.a even 1 1 trivial
560.4.a.u 3 12.b even 2 1
1225.4.a.y 3 105.g even 2 1
1575.4.a.ba 3 5.b even 2 1
2205.4.a.bm 3 7.b odd 2 1
2240.4.a.bt 3 24.h odd 2 1
2240.4.a.bv 3 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 3T_{2}^{2} - 14T_{2} + 30 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(315))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 3 T^{2} + \cdots + 30 \) 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
$11$ \( T^{3} - 74 T^{2} + \cdots - 7692 \) Copy content Toggle raw display
$13$ \( T^{3} - 44 T^{2} + \cdots - 44870 \) Copy content Toggle raw display
$17$ \( T^{3} - 52 T^{2} + \cdots + 56706 \) Copy content Toggle raw display
$19$ \( T^{3} - 168 T^{2} + \cdots - 28720 \) Copy content Toggle raw display
$23$ \( T^{3} - 124 T^{2} + \cdots + 94368 \) Copy content Toggle raw display
$29$ \( T^{3} + 332 T^{2} + \cdots - 2565450 \) Copy content Toggle raw display
$31$ \( T^{3} - 320 T^{2} + \cdots + 50176 \) Copy content Toggle raw display
$37$ \( T^{3} + 54 T^{2} + \cdots + 25736 \) Copy content Toggle raw display
$41$ \( T^{3} + 362 T^{2} + \cdots + 1536192 \) Copy content Toggle raw display
$43$ \( T^{3} + 16 T^{2} + \cdots - 1524560 \) Copy content Toggle raw display
$47$ \( T^{3} - 730 T^{2} + \cdots - 4968912 \) Copy content Toggle raw display
$53$ \( T^{3} + 110 T^{2} + \cdots - 90318336 \) Copy content Toggle raw display
$59$ \( T^{3} - 180 T^{2} + \cdots + 202459200 \) Copy content Toggle raw display
$61$ \( T^{3} - 1222 T^{2} + \cdots - 38393792 \) Copy content Toggle raw display
$67$ \( T^{3} - 204 T^{2} + \cdots + 324944128 \) Copy content Toggle raw display
$71$ \( T^{3} - 136 T^{2} + \cdots - 15575040 \) Copy content Toggle raw display
$73$ \( T^{3} - 310 T^{2} + \cdots + 48718616 \) Copy content Toggle raw display
$79$ \( T^{3} + 1034 T^{2} + \cdots - 343615600 \) Copy content Toggle raw display
$83$ \( T^{3} - 1660 T^{2} + \cdots + 42727104 \) Copy content Toggle raw display
$89$ \( T^{3} + 242 T^{2} + \cdots + 6359520 \) Copy content Toggle raw display
$97$ \( T^{3} - 100 T^{2} + \cdots - 1978018 \) Copy content Toggle raw display
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