Properties

Label 315.10.a.f
Level $315$
Weight $10$
Character orbit 315.a
Self dual yes
Analytic conductor $162.236$
Analytic rank $0$
Dimension $4$
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] = [315,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(162.236288392\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 1462x^{2} + 568x + 469504 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 5 \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 4) q^{2} + (\beta_{3} + 9 \beta_1 + 235) q^{4} - 625 q^{5} - 2401 q^{7} + (13 \beta_{3} + 4 \beta_{2} + \cdots + 5207) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 4) q^{2} + (\beta_{3} + 9 \beta_1 + 235) q^{4} - 625 q^{5} - 2401 q^{7} + (13 \beta_{3} + 4 \beta_{2} + \cdots + 5207) q^{8}+ \cdots + (5764801 \beta_1 + 23059204) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 17 q^{2} + 949 q^{4} - 2500 q^{5} - 9604 q^{7} + 20679 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 17 q^{2} + 949 q^{4} - 2500 q^{5} - 9604 q^{7} + 20679 q^{8} - 10625 q^{10} - 16382 q^{11} - 84914 q^{13} - 40817 q^{14} - 829359 q^{16} + 451528 q^{17} - 1031478 q^{19} - 593125 q^{20} - 2609948 q^{22} + 5459068 q^{23} + 1562500 q^{25} - 169794 q^{26} - 2278549 q^{28} + 4867288 q^{29} + 1098642 q^{31} - 10125105 q^{32} + 10466142 q^{34} + 6002500 q^{35} + 2110068 q^{37} + 1633028 q^{38} - 12924375 q^{40} - 1104700 q^{41} - 15322648 q^{43} - 57398124 q^{44} + 4904680 q^{46} + 5033968 q^{47} + 23059204 q^{49} + 6640625 q^{50} - 153789226 q^{52} + 149234422 q^{53} + 10238750 q^{55} - 49650279 q^{56} + 7850594 q^{58} + 141913876 q^{59} + 235578792 q^{61} - 60170880 q^{62} + 30064673 q^{64} + 53071250 q^{65} - 401097064 q^{67} + 170870006 q^{68} + 25510625 q^{70} + 126532750 q^{71} - 653180926 q^{73} + 1130189382 q^{74} + 7344404 q^{76} + 39333182 q^{77} - 624551664 q^{79} + 518349375 q^{80} + 193533606 q^{82} + 702347048 q^{83} - 282205000 q^{85} + 2076482652 q^{86} - 871546276 q^{88} + 1510406712 q^{89} + 203878514 q^{91} + 1662743304 q^{92} - 852748960 q^{94} + 644673750 q^{95} - 717560562 q^{97} + 98001617 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 1462x^{2} + 568x + 469504 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu^{2} - 818\nu + 264 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 731 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 731 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_{2} + 819\beta _1 + 467 \) 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
−30.9927
−21.7320
21.9121
31.8126
−26.9927 0 216.604 −625.000 0 −2401.00 7973.53 0 16870.4
1.2 −17.7320 0 −197.578 −625.000 0 −2401.00 12582.2 0 11082.5
1.3 25.9121 0 159.434 −625.000 0 −2401.00 −9135.70 0 −16195.0
1.4 35.8126 0 770.540 −625.000 0 −2401.00 9258.97 0 −22382.9
\(n\): 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 315.10.a.f 4
3.b odd 2 1 105.10.a.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.10.a.c 4 3.b odd 2 1
315.10.a.f 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 17T_{2}^{3} - 1354T_{2}^{2} + 11960T_{2} + 444160 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(315))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 17 T^{3} + \cdots + 444160 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T + 625)^{4} \) Copy content Toggle raw display
$7$ \( (T + 2401)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots - 63\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 19\!\cdots\!60 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 34\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots - 44\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 63\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 35\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 27\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots - 23\!\cdots\!40 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 50\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 94\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots - 15\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots - 64\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 10\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 22\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 36\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
show more
show less