Properties

Label 315.10.a.q
Level $315$
Weight $10$
Character orbit 315.a
Self dual yes
Analytic conductor $162.236$
Analytic rank $0$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3 x^{9} - 3667 x^{8} + 7323 x^{7} + 4338847 x^{6} - 6510663 x^{5} - 1903644413 x^{4} + \cdots - 2925217654100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{12}\cdot 5^{4} \)
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 a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - \beta_1 + 223) q^{4} + 625 q^{5} - 2401 q^{7} + ( - \beta_{3} - \beta_{2} - 280 \beta_1 + 464) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - \beta_1 + 223) q^{4} + 625 q^{5} - 2401 q^{7} + ( - \beta_{3} - \beta_{2} - 280 \beta_1 + 464) q^{8} + ( - 625 \beta_1 + 625) q^{10} + (\beta_{9} + \beta_{3} + 15 \beta_{2} + \cdots + 6559) q^{11}+ \cdots + ( - 5764801 \beta_1 + 5764801) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 7 q^{2} + 2227 q^{4} + 6250 q^{5} - 24010 q^{7} + 3801 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 7 q^{2} + 2227 q^{4} + 6250 q^{5} - 24010 q^{7} + 3801 q^{8} + 4375 q^{10} + 66164 q^{11} + 11494 q^{13} - 16807 q^{14} + 918867 q^{16} + 193034 q^{17} + 673344 q^{19} + 1391875 q^{20} - 1386368 q^{22} - 747586 q^{23} + 3906250 q^{25} - 10565862 q^{26} - 5347027 q^{28} + 1684046 q^{29} + 9842238 q^{31} - 4156131 q^{32} + 23016000 q^{34} - 15006250 q^{35} + 9741882 q^{37} - 12152762 q^{38} + 2375625 q^{40} + 14958856 q^{41} + 39625550 q^{43} + 74115048 q^{44} + 111188116 q^{46} + 69877502 q^{47} + 57648010 q^{49} + 2734375 q^{50} - 89168254 q^{52} + 151032896 q^{53} + 41352500 q^{55} - 9126201 q^{56} - 37012702 q^{58} + 81055712 q^{59} + 133907802 q^{61} + 541710636 q^{62} + 360737543 q^{64} + 7183750 q^{65} + 389884430 q^{67} + 770623072 q^{68} - 10504375 q^{70} + 318130274 q^{71} - 97314154 q^{73} + 569155188 q^{74} + 1120827470 q^{76} - 158859764 q^{77} - 1247170680 q^{79} + 574291875 q^{80} - 1310498994 q^{82} - 207205436 q^{83} + 120646250 q^{85} + 3002223966 q^{86} - 2070483952 q^{88} + 660264792 q^{89} - 27597094 q^{91} - 1308446244 q^{92} - 1420882918 q^{94} + 420840000 q^{95} + 253464078 q^{97} + 40353607 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3 x^{9} - 3667 x^{8} + 7323 x^{7} + 4338847 x^{6} - 6510663 x^{5} - 1903644413 x^{4} + \cdots - 2925217654100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 734 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu^{2} - 1300\nu + 2221 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1102135 \nu^{9} + 1584867766 \nu^{8} - 34410507895 \nu^{7} - 5319688467150 \nu^{6} + \cdots + 13\!\cdots\!44 ) / 183178893482496 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 7427897 \nu^{9} - 67925338 \nu^{8} + 22243759721 \nu^{7} + 373369732658 \nu^{6} + \cdots - 34\!\cdots\!12 ) / 183178893482496 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 18172435 \nu^{9} + 240505438 \nu^{8} - 63081423971 \nu^{7} - 897412069094 \nu^{6} + \cdots + 29\!\cdots\!24 ) / 183178893482496 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 19198605 \nu^{9} + 1258794766 \nu^{8} + 54472412109 \nu^{7} - 3977836407750 \nu^{6} + \cdots + 10\!\cdots\!16 ) / 183178893482496 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 7550281 \nu^{9} - 39296650 \nu^{8} - 25372278649 \nu^{7} + 89512194086 \nu^{6} + \cdots + 68\!\cdots\!24 ) / 22897361685312 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 125147307 \nu^{9} - 2023391842 \nu^{8} - 415941351051 \nu^{7} + 6156770978442 \nu^{6} + \cdots - 70\!\cdots\!80 ) / 366357786964992 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 734 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_{2} + 1304\beta _1 + 715 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{9} - 2\beta_{8} + \beta_{7} - 2\beta_{5} + 2\beta_{3} + 1837\beta_{2} + 4006\beta _1 + 957060 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 20 \beta_{9} - 4 \beta_{8} + 22 \beta_{7} - 45 \beta_{6} - 31 \beta_{5} + 2299 \beta_{3} + \cdots + 2905976 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 5934 \beta_{9} - 5382 \beta_{8} + 3615 \beta_{7} - 729 \beta_{6} - 4941 \beta_{5} - 252 \beta_{4} + \cdots + 1471481855 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 58500 \beta_{9} - 6900 \beta_{8} + 76554 \beta_{7} - 140310 \beta_{6} - 105630 \beta_{5} + \cdots + 7304887899 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13199502 \beta_{9} - 11296590 \beta_{8} + 8979135 \beta_{7} - 2503980 \beta_{6} - 10132674 \beta_{5} + \cdots + 2426673958588 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 140684604 \beta_{9} - 14781228 \beta_{8} + 193590282 \beta_{7} - 324333711 \beta_{6} + \cdots + 16513930227194 \) 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
43.2473
34.0884
22.2570
13.1091
6.43866
−1.95949
−19.3422
−20.3576
−33.0061
−41.4751
−42.2473 0 1272.83 625.000 0 −2401.00 −32143.0 0 −26404.5
1.2 −33.0884 0 582.845 625.000 0 −2401.00 −2344.15 0 −20680.3
1.3 −21.2570 0 −60.1407 625.000 0 −2401.00 12162.0 0 −13285.6
1.4 −12.1091 0 −365.369 625.000 0 −2401.00 10624.2 0 −7568.19
1.5 −5.43866 0 −482.421 625.000 0 −2401.00 5408.32 0 −3399.16
1.6 2.95949 0 −503.241 625.000 0 −2401.00 −3004.60 0 1849.68
1.7 20.3422 0 −98.1948 625.000 0 −2401.00 −12412.7 0 12713.9
1.8 21.3576 0 −55.8526 625.000 0 −2401.00 −12128.0 0 13348.5
1.9 34.0061 0 644.414 625.000 0 −2401.00 4502.90 0 21253.8
1.10 42.4751 0 1292.13 625.000 0 −2401.00 33136.1 0 26546.9
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.q yes 10
3.b odd 2 1 315.10.a.p 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.10.a.p 10 3.b odd 2 1
315.10.a.q yes 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 7 T_{2}^{9} - 3649 T_{2}^{8} + 22001 T_{2}^{7} + 4287390 T_{2}^{6} - 19470724 T_{2}^{5} + \cdots - 3634454155264 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(315))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots - 3634454155264 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T - 625)^{10} \) Copy content Toggle raw display
$7$ \( (T + 2401)^{10} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 29\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots - 17\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots - 37\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 26\!\cdots\!12 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 21\!\cdots\!72 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 13\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 15\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 30\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 77\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 17\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 20\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 44\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 24\!\cdots\!28 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 35\!\cdots\!48 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 87\!\cdots\!52 \) Copy content Toggle raw display
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