Properties

Label 315.4.a.f
Level $315$
Weight $4$
Character orbit 315.a
Self dual yes
Analytic conductor $18.586$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) 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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 4) q^{2} + ( - 8 \beta + 10) q^{4} + 5 q^{5} - 7 q^{7} + (34 \beta - 24) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 4) q^{2} + ( - 8 \beta + 10) q^{4} + 5 q^{5} - 7 q^{7} + (34 \beta - 24) q^{8} + (5 \beta - 20) q^{10} + ( - 32 \beta + 7) q^{11} + ( - 4 \beta + 25) q^{13} + ( - 7 \beta + 28) q^{14} + ( - 96 \beta + 84) q^{16} + (44 \beta + 25) q^{17} + ( - 44 \beta + 18) q^{19} + ( - 40 \beta + 50) q^{20} + (135 \beta - 92) q^{22} + ( - 68 \beta - 122) q^{23} + 25 q^{25} + (41 \beta - 108) q^{26} + (56 \beta - 70) q^{28} + (24 \beta + 13) q^{29} + (180 \beta - 60) q^{31} + (196 \beta - 336) q^{32} + ( - 151 \beta - 12) q^{34} - 35 q^{35} + (60 \beta + 282) q^{37} + (194 \beta - 160) q^{38} + (170 \beta - 120) q^{40} + (124 \beta + 164) q^{41} + ( - 68 \beta - 130) q^{43} + ( - 376 \beta + 582) q^{44} + (150 \beta + 352) q^{46} + ( - 132 \beta + 175) q^{47} + 49 q^{49} + (25 \beta - 100) q^{50} + ( - 240 \beta + 314) q^{52} + (128 \beta + 28) q^{53} + ( - 160 \beta + 35) q^{55} + ( - 238 \beta + 168) q^{56} + ( - 83 \beta - 4) q^{58} + 616 q^{59} + (108 \beta + 168) q^{61} + ( - 780 \beta + 600) q^{62} + ( - 352 \beta + 1064) q^{64} + ( - 20 \beta + 125) q^{65} + (64 \beta - 76) q^{67} + (240 \beta - 454) q^{68} + ( - 35 \beta + 140) q^{70} + 952 q^{71} + (344 \beta + 338) q^{73} + (42 \beta - 1008) q^{74} + ( - 584 \beta + 884) q^{76} + (224 \beta - 49) q^{77} + ( - 248 \beta + 507) q^{79} + ( - 480 \beta + 420) q^{80} + ( - 332 \beta - 408) q^{82} + (600 \beta + 188) q^{83} + (220 \beta + 125) q^{85} + (142 \beta + 384) q^{86} + (1006 \beta - 2344) q^{88} + (44 \beta + 108) q^{89} + (28 \beta - 175) q^{91} + (296 \beta - 132) q^{92} + (703 \beta - 964) q^{94} + ( - 220 \beta + 90) q^{95} + ( - 220 \beta + 1371) q^{97} + (49 \beta - 196) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} + 20 q^{4} + 10 q^{5} - 14 q^{7} - 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{2} + 20 q^{4} + 10 q^{5} - 14 q^{7} - 48 q^{8} - 40 q^{10} + 14 q^{11} + 50 q^{13} + 56 q^{14} + 168 q^{16} + 50 q^{17} + 36 q^{19} + 100 q^{20} - 184 q^{22} - 244 q^{23} + 50 q^{25} - 216 q^{26} - 140 q^{28} + 26 q^{29} - 120 q^{31} - 672 q^{32} - 24 q^{34} - 70 q^{35} + 564 q^{37} - 320 q^{38} - 240 q^{40} + 328 q^{41} - 260 q^{43} + 1164 q^{44} + 704 q^{46} + 350 q^{47} + 98 q^{49} - 200 q^{50} + 628 q^{52} + 56 q^{53} + 70 q^{55} + 336 q^{56} - 8 q^{58} + 1232 q^{59} + 336 q^{61} + 1200 q^{62} + 2128 q^{64} + 250 q^{65} - 152 q^{67} - 908 q^{68} + 280 q^{70} + 1904 q^{71} + 676 q^{73} - 2016 q^{74} + 1768 q^{76} - 98 q^{77} + 1014 q^{79} + 840 q^{80} - 816 q^{82} + 376 q^{83} + 250 q^{85} + 768 q^{86} - 4688 q^{88} + 216 q^{89} - 350 q^{91} - 264 q^{92} - 1928 q^{94} + 180 q^{95} + 2742 q^{97} - 392 q^{98}+O(q^{100}) \) 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
−1.41421
1.41421
−5.41421 0 21.3137 5.00000 0 −7.00000 −72.0833 0 −27.0711
1.2 −2.58579 0 −1.31371 5.00000 0 −7.00000 24.0833 0 −12.9289
\(n\): 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.f 2
3.b odd 2 1 35.4.a.b 2
5.b even 2 1 1575.4.a.z 2
7.b odd 2 1 2205.4.a.u 2
12.b even 2 1 560.4.a.r 2
15.d odd 2 1 175.4.a.c 2
15.e even 4 2 175.4.b.c 4
21.c even 2 1 245.4.a.k 2
21.g even 6 2 245.4.e.i 4
21.h odd 6 2 245.4.e.h 4
24.f even 2 1 2240.4.a.bo 2
24.h odd 2 1 2240.4.a.bn 2
105.g even 2 1 1225.4.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.b 2 3.b odd 2 1
175.4.a.c 2 15.d odd 2 1
175.4.b.c 4 15.e even 4 2
245.4.a.k 2 21.c even 2 1
245.4.e.h 4 21.h odd 6 2
245.4.e.i 4 21.g even 6 2
315.4.a.f 2 1.a even 1 1 trivial
560.4.a.r 2 12.b even 2 1
1225.4.a.m 2 105.g even 2 1
1575.4.a.z 2 5.b even 2 1
2205.4.a.u 2 7.b odd 2 1
2240.4.a.bn 2 24.h odd 2 1
2240.4.a.bo 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 8T_{2} + 14 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(315))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 8T + 14 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 5)^{2} \) Copy content Toggle raw display
$7$ \( (T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 14T - 1999 \) Copy content Toggle raw display
$13$ \( T^{2} - 50T + 593 \) Copy content Toggle raw display
$17$ \( T^{2} - 50T - 3247 \) Copy content Toggle raw display
$19$ \( T^{2} - 36T - 3548 \) Copy content Toggle raw display
$23$ \( T^{2} + 244T + 5636 \) Copy content Toggle raw display
$29$ \( T^{2} - 26T - 983 \) Copy content Toggle raw display
$31$ \( T^{2} + 120T - 61200 \) Copy content Toggle raw display
$37$ \( T^{2} - 564T + 72324 \) Copy content Toggle raw display
$41$ \( T^{2} - 328T - 3856 \) Copy content Toggle raw display
$43$ \( T^{2} + 260T + 7652 \) Copy content Toggle raw display
$47$ \( T^{2} - 350T - 4223 \) Copy content Toggle raw display
$53$ \( T^{2} - 56T - 31984 \) Copy content Toggle raw display
$59$ \( (T - 616)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 336T + 4896 \) Copy content Toggle raw display
$67$ \( T^{2} + 152T - 2416 \) Copy content Toggle raw display
$71$ \( (T - 952)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 676T - 122428 \) Copy content Toggle raw display
$79$ \( T^{2} - 1014 T + 134041 \) Copy content Toggle raw display
$83$ \( T^{2} - 376T - 684656 \) Copy content Toggle raw display
$89$ \( T^{2} - 216T + 7792 \) Copy content Toggle raw display
$97$ \( T^{2} - 2742 T + 1782841 \) Copy content Toggle raw display
show more
show less