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Properties

Label 315.4.d.b
Level $315$
Weight $4$
Character orbit 315.d
Analytic conductor $18.586$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,4,Mod(64,315)]
 
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, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.64");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 315.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(18.5856016518\)
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: \( x^{10} + 37x^{8} + 398x^{6} + 1149x^{4} + 1040x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 5 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + (\beta_{5} + \beta_1 - 5) q^{4} + ( - \beta_{8} - \beta_{6} + \beta_1 + 2) q^{5} + 7 \beta_{2} q^{7} + (\beta_{9} - 3 \beta_{8} - \beta_{7} + \cdots + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + (\beta_{5} + \beta_1 - 5) q^{4} + ( - \beta_{8} - \beta_{6} + \beta_1 + 2) q^{5} + 7 \beta_{2} q^{7} + (\beta_{9} - 3 \beta_{8} - \beta_{7} + \cdots + 1) q^{8}+ \cdots - 49 \beta_{6} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 54 q^{4} + 14 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 54 q^{4} + 14 q^{5} + 92 q^{10} - 132 q^{11} + 14 q^{14} + 310 q^{16} - 348 q^{19} - 366 q^{20} - 374 q^{25} - 892 q^{26} + 740 q^{29} + 684 q^{31} - 224 q^{34} - 2156 q^{40} - 1604 q^{41} + 580 q^{44} + 1280 q^{46} - 490 q^{49} + 2504 q^{50} - 452 q^{55} - 462 q^{56} + 1408 q^{59} + 1300 q^{61} - 150 q^{64} + 3296 q^{65} - 882 q^{70} - 2940 q^{71} - 2624 q^{74} + 8740 q^{76} + 1640 q^{79} + 4126 q^{80} - 1704 q^{85} - 1664 q^{86} + 572 q^{89} - 28 q^{91} - 5080 q^{94} - 1268 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 37x^{8} + 398x^{6} + 1149x^{4} + 1040x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 16\nu^{8} + 561\nu^{6} + 5285\nu^{4} + 8248\nu^{2} + 880 ) / 105 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 57\nu^{9} + 1999\nu^{7} + 18816\nu^{5} + 28813\nu^{3} + 790\nu ) / 700 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 73\nu^{8} + 2574\nu^{6} + 24563\nu^{4} + 41065\nu^{2} + 6535 ) / 105 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 88\nu^{8} + 3096\nu^{6} + 29414\nu^{4} + 48052\nu^{2} + 210\nu + 5995 ) / 105 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -130\nu^{8} - 4566\nu^{6} - 43148\nu^{4} - 67876\nu^{2} - 5575 ) / 105 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 157\nu^{9} + 5514\nu^{7} + 52136\nu^{5} + 82603\nu^{3} + 7690\nu ) / 525 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 949\nu^{9} + 33483\nu^{7} + 320432\nu^{5} + 548881\nu^{3} + 134830\nu ) / 2100 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1223 \nu^{9} - 660 \nu^{8} - 43161 \nu^{7} - 23220 \nu^{6} - 413224 \nu^{5} - 220080 \nu^{4} + \cdots - 34200 ) / 2100 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -1373\nu^{9} - 48591\nu^{7} - 468664\nu^{5} - 838937\nu^{3} - 228710\nu ) / 2100 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( 3\beta_{9} - 3\beta_{8} + 3\beta_{7} - 3\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 3\beta_{2} + \beta _1 + 1 ) / 20 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + 2\beta_{5} - 3\beta_{4} + 3\beta_{3} - \beta_{2} + 17\beta _1 - 68 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 26 \beta_{9} + 36 \beta_{8} - 11 \beta_{7} + 16 \beta_{6} - 12 \beta_{5} - 12 \beta_{4} + 12 \beta_{3} + \cdots - 12 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 53 \beta_{9} + 53 \beta_{8} - 53 \beta_{7} + 53 \beta_{6} - 51 \beta_{5} + 159 \beta_{4} + \cdots + 2059 ) / 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 933 \beta_{9} - 1323 \beta_{8} + 283 \beta_{7} - 133 \beta_{6} + 441 \beta_{5} + 441 \beta_{4} + \cdots + 441 ) / 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1177 \beta_{9} - 1177 \beta_{8} + 1177 \beta_{7} - 1177 \beta_{6} + 639 \beta_{5} - 3531 \beta_{4} + \cdots - 35351 ) / 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4283 \beta_{9} + 6033 \beta_{8} - 1028 \beta_{7} - 1112 \beta_{6} - 2011 \beta_{5} - 2011 \beta_{4} + \cdots - 2011 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 24793 \beta_{9} + 24793 \beta_{8} - 24793 \beta_{7} + 24793 \beta_{6} - 7621 \beta_{5} + \cdots + 628389 ) / 20 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 159539 \beta_{9} - 222969 \beta_{8} + 30934 \beta_{7} + 91881 \beta_{6} + 74323 \beta_{5} + \cdots + 74323 ) / 10 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
3.71490i
− 1.37042i
− 4.40248i
1.35311i
0.329739i
− 0.329739i
− 1.35311i
4.40248i
1.37042i
− 3.71490i
− 5.18660i 0 −18.9008 4.24321 + 10.3438i 0 7.00000i 56.5383i 0 53.6494 − 22.0078i
64.2 − 4.88936i 0 −15.9059 9.63020 + 5.67972i 0 − 7.00000i 38.6546i 0 27.7702 − 47.0855i
64.3 − 3.33774i 0 −3.14050 −10.1427 − 4.70380i 0 7.00000i − 16.2197i 0 −15.7000 + 33.8537i
64.4 − 2.20666i 0 3.13065 1.50045 − 11.0792i 0 − 7.00000i − 24.5616i 0 −24.4480 − 3.31098i
64.5 − 0.428319i 0 7.81654 1.76884 + 11.0395i 0 − 7.00000i − 6.77452i 0 4.72844 − 0.757628i
64.6 0.428319i 0 7.81654 1.76884 − 11.0395i 0 7.00000i 6.77452i 0 4.72844 + 0.757628i
64.7 2.20666i 0 3.13065 1.50045 + 11.0792i 0 7.00000i 24.5616i 0 −24.4480 + 3.31098i
64.8 3.33774i 0 −3.14050 −10.1427 + 4.70380i 0 − 7.00000i 16.2197i 0 −15.7000 − 33.8537i
64.9 4.88936i 0 −15.9059 9.63020 − 5.67972i 0 7.00000i − 38.6546i 0 27.7702 + 47.0855i
64.10 5.18660i 0 −18.9008 4.24321 − 10.3438i 0 − 7.00000i − 56.5383i 0 53.6494 + 22.0078i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.4.d.b 10
3.b odd 2 1 105.4.d.b ✓ 10
5.b even 2 1 inner 315.4.d.b 10
5.c odd 4 1 1575.4.a.bo 5
5.c odd 4 1 1575.4.a.bp 5
15.d odd 2 1 105.4.d.b ✓ 10
15.e even 4 1 525.4.a.w 5
15.e even 4 1 525.4.a.x 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.d.b ✓ 10 3.b odd 2 1
105.4.d.b ✓ 10 15.d odd 2 1
315.4.d.b 10 1.a even 1 1 trivial
315.4.d.b 10 5.b even 2 1 inner
525.4.a.w 5 15.e even 4 1
525.4.a.x 5 15.e even 4 1
1575.4.a.bo 5 5.c odd 4 1
1575.4.a.bp 5 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 67T_{2}^{8} + 1523T_{2}^{6} + 13329T_{2}^{4} + 37280T_{2}^{2} + 6400 \) acting on \(S_{4}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 67 T^{8} + \cdots + 6400 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 30517578125 \) Copy content Toggle raw display
$7$ \( (T^{2} + 49)^{5} \) Copy content Toggle raw display
$11$ \( (T^{5} + 66 T^{4} + \cdots + 55852416)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 53\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( (T^{5} + 174 T^{4} + \cdots + 784374624)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{5} - 370 T^{4} + \cdots + 1150048)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 342 T^{4} + \cdots + 52737095200)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 99\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( (T^{5} + 802 T^{4} + \cdots - 531402107648)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 21\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots - 33724261457920)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 1105143174112)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 24\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( (T^{5} + 1470 T^{4} + \cdots - 237519904000)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 37\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 43229481181184)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 1125486676224)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 39\!\cdots\!56 \) Copy content Toggle raw display
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