Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + (\beta - 4) q^{4} - 5 q^{5} - 7 q^{7} + (11 \beta - 4) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + (\beta - 4) q^{4} - 5 q^{5} - 7 q^{7} + (11 \beta - 4) q^{8} + 5 \beta q^{10} + ( - 4 \beta + 4) q^{11} - 22 \beta q^{13} + 7 \beta q^{14} + ( - 15 \beta - 12) q^{16} + ( - 26 \beta + 42) q^{17} + ( - 44 \beta + 22) q^{19} + ( - 5 \beta + 20) q^{20} + 16 q^{22} + (14 \beta + 34) q^{23} + 25 q^{25} + (22 \beta + 88) q^{26} + ( - 7 \beta + 28) q^{28} + (30 \beta + 152) q^{29} + (18 \beta - 114) q^{31} + ( - 61 \beta + 92) q^{32} + ( - 16 \beta + 104) q^{34} + 35 q^{35} + ( - 30 \beta + 18) q^{37} + (22 \beta + 176) q^{38} + ( - 55 \beta + 20) q^{40} + ( - 52 \beta + 114) q^{41} + (166 \beta - 60) q^{43} + (16 \beta - 32) q^{44} + ( - 48 \beta - 56) q^{46} + ( - 30 \beta + 272) q^{47} + 49 q^{49} - 25 \beta q^{50} + (66 \beta - 88) q^{52} + (52 \beta + 378) q^{53} + (20 \beta - 20) q^{55} + ( - 77 \beta + 28) q^{56} + ( - 182 \beta - 120) q^{58} + 284 q^{59} + (90 \beta - 354) q^{61} + (96 \beta - 72) q^{62} + (89 \beta + 340) q^{64} + 110 \beta q^{65} + ( - 98 \beta + 396) q^{67} + (120 \beta - 272) q^{68} - 35 \beta q^{70} + ( - 162 \beta + 488) q^{71} + (290 \beta - 104) q^{73} + (12 \beta + 120) q^{74} + (154 \beta - 264) q^{76} + (28 \beta - 28) q^{77} + (328 \beta + 136) q^{79} + (75 \beta + 60) q^{80} + ( - 62 \beta + 208) q^{82} + (300 \beta - 284) q^{83} + (130 \beta - 210) q^{85} + ( - 106 \beta - 664) q^{86} + (16 \beta - 192) q^{88} + ( - 476 \beta + 202) q^{89} + 154 \beta q^{91} + ( - 8 \beta - 80) q^{92} + ( - 242 \beta + 120) q^{94} + (220 \beta - 110) q^{95} + (482 \beta + 572) q^{97} - 49 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 7 q^{4} - 10 q^{5} - 14 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 7 q^{4} - 10 q^{5} - 14 q^{7} + 3 q^{8} + 5 q^{10} + 4 q^{11} - 22 q^{13} + 7 q^{14} - 39 q^{16} + 58 q^{17} + 35 q^{20} + 32 q^{22} + 82 q^{23} + 50 q^{25} + 198 q^{26} + 49 q^{28} + 334 q^{29} - 210 q^{31} + 123 q^{32} + 192 q^{34} + 70 q^{35} + 6 q^{37} + 374 q^{38} - 15 q^{40} + 176 q^{41} + 46 q^{43} - 48 q^{44} - 160 q^{46} + 514 q^{47} + 98 q^{49} - 25 q^{50} - 110 q^{52} + 808 q^{53} - 20 q^{55} - 21 q^{56} - 422 q^{58} + 568 q^{59} - 618 q^{61} - 48 q^{62} + 769 q^{64} + 110 q^{65} + 694 q^{67} - 424 q^{68} - 35 q^{70} + 814 q^{71} + 82 q^{73} + 252 q^{74} - 374 q^{76} - 28 q^{77} + 600 q^{79} + 195 q^{80} + 354 q^{82} - 268 q^{83} - 290 q^{85} - 1434 q^{86} - 368 q^{88} - 72 q^{89} + 154 q^{91} - 168 q^{92} - 2 q^{94} + 1626 q^{97} - 49 q^{98}+O(q^{100}) \) 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
2.56155
−1.56155
−2.56155 0 −1.43845 −5.00000 0 −7.00000 24.1771 0 12.8078
1.2 1.56155 0 −5.56155 −5.00000 0 −7.00000 −21.1771 0 −7.80776
\(n\): 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.h 2
3.b odd 2 1 315.4.a.j yes 2
5.b even 2 1 1575.4.a.u 2
7.b odd 2 1 2205.4.a.y 2
15.d odd 2 1 1575.4.a.r 2
21.c even 2 1 2205.4.a.ba 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.4.a.h 2 1.a even 1 1 trivial
315.4.a.j yes 2 3.b odd 2 1
1575.4.a.r 2 15.d odd 2 1
1575.4.a.u 2 5.b even 2 1
2205.4.a.y 2 7.b odd 2 1
2205.4.a.ba 2 21.c even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 4 \) 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
$11$ \( T^{2} - 4T - 64 \) Copy content Toggle raw display
$13$ \( T^{2} + 22T - 1936 \) Copy content Toggle raw display
$17$ \( T^{2} - 58T - 2032 \) Copy content Toggle raw display
$19$ \( T^{2} - 8228 \) Copy content Toggle raw display
$23$ \( T^{2} - 82T + 848 \) Copy content Toggle raw display
$29$ \( T^{2} - 334T + 24064 \) Copy content Toggle raw display
$31$ \( T^{2} + 210T + 9648 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T - 3816 \) Copy content Toggle raw display
$41$ \( T^{2} - 176T - 3748 \) Copy content Toggle raw display
$43$ \( T^{2} - 46T - 116584 \) Copy content Toggle raw display
$47$ \( T^{2} - 514T + 62224 \) Copy content Toggle raw display
$53$ \( T^{2} - 808T + 151724 \) Copy content Toggle raw display
$59$ \( (T - 284)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 618T + 61056 \) Copy content Toggle raw display
$67$ \( T^{2} - 694T + 79592 \) Copy content Toggle raw display
$71$ \( T^{2} - 814T + 54112 \) Copy content Toggle raw display
$73$ \( T^{2} - 82T - 355744 \) Copy content Toggle raw display
$79$ \( T^{2} - 600T - 367232 \) Copy content Toggle raw display
$83$ \( T^{2} + 268T - 364544 \) Copy content Toggle raw display
$89$ \( T^{2} + 72T - 961652 \) Copy content Toggle raw display
$97$ \( T^{2} - 1626 T - 326408 \) Copy content Toggle raw display
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