Properties

Label 315.4.a.g
Level $315$
Weight $4$
Character orbit 315.a
Self dual yes
Analytic conductor $18.586$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} + (3 \beta + 3) q^{4} + 5 q^{5} + 7 q^{7} + ( - \beta - 25) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} + (3 \beta + 3) q^{4} + 5 q^{5} + 7 q^{7} + ( - \beta - 25) q^{8} + ( - 5 \beta - 5) q^{10} + (2 \beta - 32) q^{11} + ( - 10 \beta + 2) q^{13} + ( - 7 \beta - 7) q^{14} + (3 \beta + 11) q^{16} + (12 \beta - 26) q^{17} + ( - 2 \beta - 60) q^{19} + (15 \beta + 15) q^{20} + (28 \beta + 12) q^{22} + (48 \beta - 32) q^{23} + 25 q^{25} + (18 \beta + 98) q^{26} + (21 \beta + 21) q^{28} + ( - 12 \beta - 170) q^{29} + ( - 38 \beta + 52) q^{31} + ( - 9 \beta + 159) q^{32} + (2 \beta - 94) q^{34} + 35 q^{35} + (80 \beta - 134) q^{37} + (64 \beta + 80) q^{38} + ( - 5 \beta - 125) q^{40} + (108 \beta - 62) q^{41} + (100 \beta - 248) q^{43} + ( - 84 \beta - 36) q^{44} + ( - 64 \beta - 448) q^{46} + ( - 140 \beta + 164) q^{47} + 49 q^{49} + ( - 25 \beta - 25) q^{50} + ( - 54 \beta - 294) q^{52} + ( - 58 \beta - 462) q^{53} + (10 \beta - 160) q^{55} + ( - 7 \beta - 175) q^{56} + (194 \beta + 290) q^{58} + ( - 76 \beta - 220) q^{59} + ( - 84 \beta - 398) q^{61} + (24 \beta + 328) q^{62} + ( - 165 \beta - 157) q^{64} + ( - 50 \beta + 10) q^{65} + ( - 228 \beta - 64) q^{67} + ( - 6 \beta + 282) q^{68} + ( - 35 \beta - 35) q^{70} + ( - 86 \beta - 112) q^{71} + ( - 38 \beta + 182) q^{73} + ( - 26 \beta - 666) q^{74} + ( - 192 \beta - 240) q^{76} + (14 \beta - 224) q^{77} + ( - 8 \beta + 920) q^{79} + (15 \beta + 55) q^{80} + ( - 154 \beta - 1018) q^{82} + (224 \beta + 228) q^{83} + (60 \beta - 130) q^{85} + (48 \beta - 752) q^{86} + ( - 20 \beta + 780) q^{88} + ( - 336 \beta - 230) q^{89} + ( - 70 \beta + 14) q^{91} + (192 \beta + 1344) q^{92} + (116 \beta + 1236) q^{94} + ( - 10 \beta - 300) q^{95} + (278 \beta - 474) q^{97} + ( - 49 \beta - 49) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 9 q^{4} + 10 q^{5} + 14 q^{7} - 51 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 9 q^{4} + 10 q^{5} + 14 q^{7} - 51 q^{8} - 15 q^{10} - 62 q^{11} - 6 q^{13} - 21 q^{14} + 25 q^{16} - 40 q^{17} - 122 q^{19} + 45 q^{20} + 52 q^{22} - 16 q^{23} + 50 q^{25} + 214 q^{26} + 63 q^{28} - 352 q^{29} + 66 q^{31} + 309 q^{32} - 186 q^{34} + 70 q^{35} - 188 q^{37} + 224 q^{38} - 255 q^{40} - 16 q^{41} - 396 q^{43} - 156 q^{44} - 960 q^{46} + 188 q^{47} + 98 q^{49} - 75 q^{50} - 642 q^{52} - 982 q^{53} - 310 q^{55} - 357 q^{56} + 774 q^{58} - 516 q^{59} - 880 q^{61} + 680 q^{62} - 479 q^{64} - 30 q^{65} - 356 q^{67} + 558 q^{68} - 105 q^{70} - 310 q^{71} + 326 q^{73} - 1358 q^{74} - 672 q^{76} - 434 q^{77} + 1832 q^{79} + 125 q^{80} - 2190 q^{82} + 680 q^{83} - 200 q^{85} - 1456 q^{86} + 1540 q^{88} - 796 q^{89} - 42 q^{91} + 2880 q^{92} + 2588 q^{94} - 610 q^{95} - 670 q^{97} - 147 q^{98}+O(q^{100}) \) 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
3.70156
−2.70156
−4.70156 0 14.1047 5.00000 0 7.00000 −28.7016 0 −23.5078
1.2 1.70156 0 −5.10469 5.00000 0 7.00000 −22.2984 0 8.50781
\(n\): 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.g 2
3.b odd 2 1 105.4.a.g 2
5.b even 2 1 1575.4.a.y 2
7.b odd 2 1 2205.4.a.v 2
12.b even 2 1 1680.4.a.y 2
15.d odd 2 1 525.4.a.i 2
15.e even 4 2 525.4.d.j 4
21.c even 2 1 735.4.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.g 2 3.b odd 2 1
315.4.a.g 2 1.a even 1 1 trivial
525.4.a.i 2 15.d odd 2 1
525.4.d.j 4 15.e even 4 2
735.4.a.q 2 21.c even 2 1
1575.4.a.y 2 5.b even 2 1
1680.4.a.y 2 12.b even 2 1
2205.4.a.v 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3T_{2} - 8 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(315))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T - 8 \) 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
$11$ \( T^{2} + 62T + 920 \) Copy content Toggle raw display
$13$ \( T^{2} + 6T - 1016 \) Copy content Toggle raw display
$17$ \( T^{2} + 40T - 1076 \) Copy content Toggle raw display
$19$ \( T^{2} + 122T + 3680 \) Copy content Toggle raw display
$23$ \( T^{2} + 16T - 23552 \) Copy content Toggle raw display
$29$ \( T^{2} + 352T + 29500 \) Copy content Toggle raw display
$31$ \( T^{2} - 66T - 13712 \) Copy content Toggle raw display
$37$ \( T^{2} + 188T - 56764 \) Copy content Toggle raw display
$41$ \( T^{2} + 16T - 119492 \) Copy content Toggle raw display
$43$ \( T^{2} + 396T - 63296 \) Copy content Toggle raw display
$47$ \( T^{2} - 188T - 192064 \) Copy content Toggle raw display
$53$ \( T^{2} + 982T + 206600 \) Copy content Toggle raw display
$59$ \( T^{2} + 516T + 7360 \) Copy content Toggle raw display
$61$ \( T^{2} + 880T + 121276 \) Copy content Toggle raw display
$67$ \( T^{2} + 356T - 501152 \) Copy content Toggle raw display
$71$ \( T^{2} + 310T - 51784 \) Copy content Toggle raw display
$73$ \( T^{2} - 326T + 11768 \) Copy content Toggle raw display
$79$ \( T^{2} - 1832 T + 838400 \) Copy content Toggle raw display
$83$ \( T^{2} - 680T - 398704 \) Copy content Toggle raw display
$89$ \( T^{2} + 796T - 998780 \) Copy content Toggle raw display
$97$ \( T^{2} + 670T - 679936 \) Copy content Toggle raw display
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