Properties

Label 315.4.a.i
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: \(\Q(\sqrt{65}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 16 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{65})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + (\beta + 8) q^{4} - 5 q^{5} - 7 q^{7} + ( - \beta - 16) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + (\beta + 8) q^{4} - 5 q^{5} - 7 q^{7} + ( - \beta - 16) q^{8} + 5 \beta q^{10} + (2 \beta + 10) q^{11} + (2 \beta - 12) q^{13} + 7 \beta q^{14} + (9 \beta - 48) q^{16} + (16 \beta - 66) q^{17} + ( - 14 \beta + 58) q^{19} + ( - 5 \beta - 40) q^{20} + ( - 12 \beta - 32) q^{22} + (20 \beta - 140) q^{23} + 25 q^{25} + (10 \beta - 32) q^{26} + ( - 7 \beta - 56) q^{28} + (48 \beta + 74) q^{29} + (42 \beta + 54) q^{31} + (47 \beta - 16) q^{32} + (50 \beta - 256) q^{34} + 35 q^{35} + ( - 36 \beta - 30) q^{37} + ( - 44 \beta + 224) q^{38} + (5 \beta + 80) q^{40} + ( - 100 \beta + 138) q^{41} + ( - 32 \beta - 156) q^{43} + (28 \beta + 112) q^{44} + (120 \beta - 320) q^{46} + (48 \beta - 304) q^{47} + 49 q^{49} - 25 \beta q^{50} + (6 \beta - 64) q^{52} + ( - 86 \beta - 120) q^{53} + ( - 10 \beta - 50) q^{55} + (7 \beta + 112) q^{56} + ( - 122 \beta - 768) q^{58} + ( - 84 \beta + 464) q^{59} + (24 \beta - 114) q^{61} + ( - 96 \beta - 672) q^{62} + ( - 103 \beta - 368) q^{64} + ( - 10 \beta + 60) q^{65} + (64 \beta - 84) q^{67} + (78 \beta - 272) q^{68} - 35 \beta q^{70} + ( - 42 \beta - 814) q^{71} + ( - 202 \beta - 92) q^{73} + (66 \beta + 576) q^{74} + ( - 68 \beta + 240) q^{76} + ( - 14 \beta - 70) q^{77} + ( - 104 \beta - 392) q^{79} + ( - 45 \beta + 240) q^{80} + ( - 38 \beta + 1600) q^{82} + ( - 216 \beta - 356) q^{83} + ( - 80 \beta + 330) q^{85} + (188 \beta + 512) q^{86} + ( - 44 \beta - 192) q^{88} + ( - 8 \beta - 290) q^{89} + ( - 14 \beta + 84) q^{91} + (40 \beta - 800) q^{92} + (256 \beta - 768) q^{94} + (70 \beta - 290) q^{95} + (314 \beta + 104) q^{97} - 49 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 17 q^{4} - 10 q^{5} - 14 q^{7} - 33 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 17 q^{4} - 10 q^{5} - 14 q^{7} - 33 q^{8} + 5 q^{10} + 22 q^{11} - 22 q^{13} + 7 q^{14} - 87 q^{16} - 116 q^{17} + 102 q^{19} - 85 q^{20} - 76 q^{22} - 260 q^{23} + 50 q^{25} - 54 q^{26} - 119 q^{28} + 196 q^{29} + 150 q^{31} + 15 q^{32} - 462 q^{34} + 70 q^{35} - 96 q^{37} + 404 q^{38} + 165 q^{40} + 176 q^{41} - 344 q^{43} + 252 q^{44} - 520 q^{46} - 560 q^{47} + 98 q^{49} - 25 q^{50} - 122 q^{52} - 326 q^{53} - 110 q^{55} + 231 q^{56} - 1658 q^{58} + 844 q^{59} - 204 q^{61} - 1440 q^{62} - 839 q^{64} + 110 q^{65} - 104 q^{67} - 466 q^{68} - 35 q^{70} - 1670 q^{71} - 386 q^{73} + 1218 q^{74} + 412 q^{76} - 154 q^{77} - 888 q^{79} + 435 q^{80} + 3162 q^{82} - 928 q^{83} + 580 q^{85} + 1212 q^{86} - 428 q^{88} - 588 q^{89} + 154 q^{91} - 1560 q^{92} - 1280 q^{94} - 510 q^{95} + 522 q^{97} - 49 q^{98}+O(q^{100}) \) 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
4.53113
−3.53113
−4.53113 0 12.5311 −5.00000 0 −7.00000 −20.5311 0 22.6556
1.2 3.53113 0 4.46887 −5.00000 0 −7.00000 −12.4689 0 −17.6556
\(n\): 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.i 2
3.b odd 2 1 105.4.a.f 2
5.b even 2 1 1575.4.a.w 2
7.b odd 2 1 2205.4.a.z 2
12.b even 2 1 1680.4.a.bg 2
15.d odd 2 1 525.4.a.k 2
15.e even 4 2 525.4.d.h 4
21.c even 2 1 735.4.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.f 2 3.b odd 2 1
315.4.a.i 2 1.a even 1 1 trivial
525.4.a.k 2 15.d odd 2 1
525.4.d.h 4 15.e even 4 2
735.4.a.p 2 21.c even 2 1
1575.4.a.w 2 5.b even 2 1
1680.4.a.bg 2 12.b even 2 1
2205.4.a.z 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} - 16 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(315))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 16 \) 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
$11$ \( T^{2} - 22T + 56 \) Copy content Toggle raw display
$13$ \( T^{2} + 22T + 56 \) Copy content Toggle raw display
$17$ \( T^{2} + 116T - 796 \) Copy content Toggle raw display
$19$ \( T^{2} - 102T - 584 \) Copy content Toggle raw display
$23$ \( T^{2} + 260T + 10400 \) Copy content Toggle raw display
$29$ \( T^{2} - 196T - 27836 \) Copy content Toggle raw display
$31$ \( T^{2} - 150T - 23040 \) Copy content Toggle raw display
$37$ \( T^{2} + 96T - 18756 \) Copy content Toggle raw display
$41$ \( T^{2} - 176T - 154756 \) Copy content Toggle raw display
$43$ \( T^{2} + 344T + 12944 \) Copy content Toggle raw display
$47$ \( T^{2} + 560T + 40960 \) Copy content Toggle raw display
$53$ \( T^{2} + 326T - 93616 \) Copy content Toggle raw display
$59$ \( T^{2} - 844T + 63424 \) Copy content Toggle raw display
$61$ \( T^{2} + 204T + 1044 \) Copy content Toggle raw display
$67$ \( T^{2} + 104T - 63856 \) Copy content Toggle raw display
$71$ \( T^{2} + 1670 T + 668560 \) Copy content Toggle raw display
$73$ \( T^{2} + 386T - 625816 \) Copy content Toggle raw display
$79$ \( T^{2} + 888T + 21376 \) Copy content Toggle raw display
$83$ \( T^{2} + 928T - 542864 \) Copy content Toggle raw display
$89$ \( T^{2} + 588T + 85396 \) Copy content Toggle raw display
$97$ \( T^{2} - 522 T - 1534064 \) Copy content Toggle raw display
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