Properties

Label 315.10.a.l
Level $315$
Weight $10$
Character orbit 315.a
Self dual yes
Analytic conductor $162.236$
Analytic rank $0$
Dimension $6$
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] = [315,10,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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 3018x^{4} + 3368x^{3} + 2066979x^{2} - 6329061x - 14714266 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{4}\cdot 5 \)
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 3) q^{2} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 503) q^{4} - 625 q^{5} - 2401 q^{7} + ( - 5 \beta_{5} + \beta_{4} + \cdots - 3993) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 3) q^{2} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 503) q^{4} - 625 q^{5} - 2401 q^{7} + ( - 5 \beta_{5} + \beta_{4} + \cdots - 3993) q^{8}+ \cdots + (5764801 \beta_1 - 17294403) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 15 q^{2} + 3009 q^{4} - 3750 q^{5} - 14406 q^{7} - 22041 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 15 q^{2} + 3009 q^{4} - 3750 q^{5} - 14406 q^{7} - 22041 q^{8} + 9375 q^{10} + 47796 q^{11} + 102168 q^{13} + 36015 q^{14} + 2371065 q^{16} + 38472 q^{17} + 361056 q^{19} - 1880625 q^{20} + 2068680 q^{22} - 697032 q^{23} + 2343750 q^{25} - 10847622 q^{26} - 7224609 q^{28} - 16028184 q^{29} + 1362912 q^{31} + 1148823 q^{32} - 9216294 q^{34} + 9003750 q^{35} - 3912924 q^{37} - 39877140 q^{38} + 13775625 q^{40} - 22452756 q^{41} - 29998992 q^{43} + 94378728 q^{44} + 49589520 q^{46} + 121271508 q^{47} + 34588806 q^{49} - 5859375 q^{50} - 44528550 q^{52} + 20308596 q^{53} - 29872500 q^{55} + 52920441 q^{56} + 256759494 q^{58} - 120280392 q^{59} - 87693540 q^{61} + 344802528 q^{62} + 640647105 q^{64} - 63855000 q^{65} + 495050664 q^{67} + 141443706 q^{68} - 22509375 q^{70} - 253762512 q^{71} + 187195308 q^{73} + 418806054 q^{74} - 584719476 q^{76} - 114758196 q^{77} + 831079500 q^{79} - 1481915625 q^{80} + 2286848814 q^{82} + 767650536 q^{83} - 24045000 q^{85} - 566199948 q^{86} + 2952495444 q^{88} - 582579684 q^{89} - 245305368 q^{91} + 4259382384 q^{92} - 1115946660 q^{94} - 225660000 q^{95} - 1184506872 q^{97} - 86472015 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} - 3018x^{4} + 3368x^{3} + 2066979x^{2} - 6329061x - 14714266 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 71\nu^{5} - 1106\nu^{4} - 175920\nu^{3} + 1251392\nu^{2} + 92091141\nu + 14392190 ) / 520728 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 71\nu^{5} - 1106\nu^{4} - 175920\nu^{3} + 1772120\nu^{2} + 90528957\nu - 509460178 ) / 520728 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 185\nu^{5} + 3230\nu^{4} - 628292\nu^{3} - 6470588\nu^{2} + 460995047\nu + 11708850 ) / 520728 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 37\nu^{5} + 646\nu^{4} - 229804\nu^{3} - 148516\nu^{2} + 262060483\nu - 940175910 ) / 520728 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} + 3\beta _1 + 1006 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -5\beta_{5} + \beta_{4} + 11\beta_{3} - 11\beta_{2} + 1664\beta _1 + 2016 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -139\beta_{5} + 113\beta_{4} + 1898\beta_{3} - 2120\beta_{2} + 14870\beta _1 + 1662018 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -14554\beta_{5} + 4238\beta_{4} + 39196\beta_{3} - 35320\beta_{2} + 3004673\beta _1 + 12951466 \) 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
−39.7818
−36.8299
−1.54749
4.69340
29.3435
47.1222
−42.7818 0 1318.28 −625.000 0 −2401.00 −34494.1 0 26738.6
1.2 −39.8299 0 1074.42 −625.000 0 −2401.00 −22401.1 0 24893.7
1.3 −4.54749 0 −491.320 −625.000 0 −2401.00 4562.59 0 2842.18
1.4 1.69340 0 −509.132 −625.000 0 −2401.00 −1729.19 0 −1058.38
1.5 26.3435 0 181.982 −625.000 0 −2401.00 −8693.85 0 −16464.7
1.6 44.1222 0 1434.77 −625.000 0 −2401.00 40714.7 0 −27576.4
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.l 6
3.b odd 2 1 35.10.a.e 6
15.d odd 2 1 175.10.a.g 6
15.e even 4 2 175.10.b.g 12
21.c even 2 1 245.10.a.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.10.a.e 6 3.b odd 2 1
175.10.a.g 6 15.d odd 2 1
175.10.b.g 12 15.e even 4 2
245.10.a.g 6 21.c even 2 1
315.10.a.l 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 15T_{2}^{5} - 2928T_{2}^{4} - 32578T_{2}^{3} + 1934724T_{2}^{2} + 5838048T_{2} - 15252160 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(315))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 15 T^{5} + \cdots - 15252160 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T + 625)^{6} \) Copy content Toggle raw display
$7$ \( (T + 2401)^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 78\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 11\!\cdots\!60 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 10\!\cdots\!28 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots - 24\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots - 52\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots - 48\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots - 42\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 74\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 56\!\cdots\!80 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots - 15\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots - 85\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 37\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 52\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots - 68\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 20\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots - 49\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 74\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 27\!\cdots\!72 \) Copy content Toggle raw display
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