↑

Properties

Label 315.4.b.a
Level $315$
Weight $4$
Character orbit 315.b
Analytic conductor $18.586$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,4,Mod(251,315)]
 
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([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.251");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 315.b (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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 96x^{14} + 3663x^{12} + 70432x^{10} + 711687x^{8} + 3549984x^{6} + 6929305x^{4} + 543120x^{2} + 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{6} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 4) q^{4} - 5 q^{5} + (\beta_{10} + \beta_{8} - \beta_1 + 2) q^{7} + (\beta_{10} + \beta_{9} - 4 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 4) q^{4} - 5 q^{5} + (\beta_{10} + \beta_{8} - \beta_1 + 2) q^{7} + (\beta_{10} + \beta_{9} - 4 \beta_1) q^{8} - 5 \beta_1 q^{10} + ( - \beta_{11} - 3 \beta_1) q^{11} + ( - \beta_{14} + \beta_{13} + \cdots + 2 \beta_1) q^{13}+ \cdots + (4 \beta_{15} + 8 \beta_{14} + \cdots - 71) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 64 q^{4} - 80 q^{5} + 32 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 64 q^{4} - 80 q^{5} + 32 q^{7} + 108 q^{14} + 196 q^{16} + 320 q^{20} + 516 q^{22} + 400 q^{25} - 384 q^{26} - 404 q^{28} - 160 q^{35} + 952 q^{37} + 1392 q^{38} + 360 q^{41} + 424 q^{43} - 540 q^{46} - 888 q^{47} + 220 q^{49} + 12 q^{56} + 180 q^{58} + 1104 q^{59} + 3720 q^{62} + 3488 q^{64} - 2128 q^{67} - 5352 q^{68} - 540 q^{70} - 252 q^{77} - 328 q^{79} - 980 q^{80} + 2352 q^{83} - 300 q^{88} - 2616 q^{89} - 2100 q^{91} - 828 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 96x^{14} + 3663x^{12} + 70432x^{10} + 711687x^{8} + 3549984x^{6} + 6929305x^{4} + 543120x^{2} + 576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 157 \nu^{14} - 13595 \nu^{12} - 454336 \nu^{10} - 7288288 \nu^{8} - 56048491 \nu^{6} + \cdots - 5248832 ) / 5980160 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19 \nu^{14} + 1861 \nu^{12} + 73344 \nu^{10} + 1478208 \nu^{8} + 15885765 \nu^{6} + 84869283 \nu^{4} + \cdots + 3176896 ) / 373760 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 581 \nu^{14} - 54387 \nu^{12} - 1991232 \nu^{10} - 35749728 \nu^{8} - 321165699 \nu^{6} + \cdots - 18207040 ) / 2990080 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 489 \nu^{14} - 47343 \nu^{12} - 1824448 \nu^{10} - 35490912 \nu^{8} - 363254271 \nu^{6} + \cdots + 41393600 ) / 2990080 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2149 \nu^{15} - 831 \nu^{14} - 206739 \nu^{13} - 78505 \nu^{12} - 7908800 \nu^{11} + \cdots - 248381376 ) / 11960320 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2149 \nu^{15} - 831 \nu^{14} + 206739 \nu^{13} - 78505 \nu^{12} + 7908800 \nu^{11} + \cdots - 248381376 ) / 11960320 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3809 \nu^{15} + 366135 \nu^{13} + 13993152 \nu^{11} + 269638496 \nu^{9} + 2732680647 \nu^{7} + \cdots + 2708910144 \nu ) / 17940480 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3809 \nu^{15} - 366135 \nu^{13} - 13993152 \nu^{11} - 269638496 \nu^{9} + \cdots - 2350100544 \nu ) / 17940480 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 28897 \nu^{15} + 2775735 \nu^{13} + 106001856 \nu^{11} + 2040696928 \nu^{9} + \cdots + 12164292672 \nu ) / 17940480 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15393 \nu^{15} + 1478135 \nu^{13} + 56426944 \nu^{11} + 1085842272 \nu^{9} + 10986771079 \nu^{7} + \cdots + 9411631168 \nu ) / 5980160 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 16883 \nu^{15} - 659 \nu^{14} + 1620757 \nu^{13} - 64117 \nu^{12} + 61846208 \nu^{11} + \cdots - 216769728 ) / 5980160 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 104249 \nu^{15} - 1977 \nu^{14} - 10007007 \nu^{13} - 192351 \nu^{12} - 381784896 \nu^{11} + \cdots - 650309184 ) / 17940480 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 104249 \nu^{15} - 1977 \nu^{14} + 10007007 \nu^{13} - 192351 \nu^{12} + 381784896 \nu^{11} + \cdots - 650309184 ) / 17940480 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{9} - 20\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} - 25\beta_{2} + 237 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{12} + \beta_{11} - 46\beta_{10} - 33\beta_{9} - 5\beta_{8} + 5\beta_{7} + 443\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 7 \beta_{15} - 7 \beta_{14} + 42 \beta_{8} + 42 \beta_{7} - 32 \beta_{6} - \beta_{5} - 44 \beta_{4} + \cdots - 5165 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 6 \beta_{15} + 4 \beta_{14} - 10 \beta_{13} + 61 \beta_{12} - 59 \beta_{11} + 1529 \beta_{10} + \cdots - 10264 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 379 \beta_{15} + 379 \beta_{14} - 1429 \beta_{8} - 1429 \beta_{7} + 835 \beta_{6} + 77 \beta_{5} + \cdots + 117892 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 446 \beta_{15} - 292 \beta_{14} + 738 \beta_{13} - 2508 \beta_{12} + 2274 \beta_{11} + \cdots + 243573 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 14436 \beta_{15} - 14436 \beta_{14} + 44712 \beta_{8} + 44712 \beta_{7} - 20560 \beta_{6} + \cdots - 2762704 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 20616 \beta_{15} + 13584 \beta_{14} - 34200 \beta_{13} + 87932 \beta_{12} - 74084 \beta_{11} + \cdots - 5871084 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 476580 \beta_{15} + 476580 \beta_{14} - 1331433 \beta_{8} - 1331433 \beta_{7} + 495537 \beta_{6} + \cdots + 65911897 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 773448 \beta_{15} - 516176 \beta_{14} + 1289624 \beta_{13} - 2831197 \beta_{12} + \cdots + 143158555 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 14587467 \beta_{15} - 14587467 \beta_{14} + 38340330 \beta_{8} + 38340330 \beta_{7} + \cdots - 1593932561 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 25877198 \beta_{15} + 17514004 \beta_{14} - 43391202 \beta_{13} + 86480529 \beta_{12} + \cdots - 3522618592 \beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
− 5.09529i
− 4.78031i
− 4.46361i
− 3.85291i
− 2.56798i
− 2.39619i
− 0.283947i
− 0.0327915i
0.0327915i
0.283947i
2.39619i
2.56798i
3.85291i
4.46361i
4.78031i
5.09529i
− 5.09529i 0 −17.9620 −5.00000 0 −10.7370 + 15.0903i 50.7591i 0 25.4764i
251.2 − 4.78031i 0 −14.8513 −5.00000 0 18.0094 + 4.32008i 32.7514i 0 23.9015i
251.3 − 4.46361i 0 −11.9238 −5.00000 0 13.0368 − 13.1545i 17.5145i 0 22.3181i
251.4 − 3.85291i 0 −6.84495 −5.00000 0 −11.0192 − 14.8855i − 4.45031i 0 19.2646i
251.5 − 2.56798i 0 1.40549 −5.00000 0 16.0553 + 9.23194i − 24.1531i 0 12.8399i
251.6 − 2.39619i 0 2.25829 −5.00000 0 −0.977216 + 18.4945i − 24.5808i 0 11.9809i
251.7 − 0.283947i 0 7.91937 −5.00000 0 9.99806 + 15.5897i − 4.52026i 0 1.41974i
251.8 − 0.0327915i 0 7.99892 −5.00000 0 −18.3661 + 2.38487i − 0.524629i 0 0.163958i
251.9 0.0327915i 0 7.99892 −5.00000 0 −18.3661 − 2.38487i 0.524629i 0 − 0.163958i
251.10 0.283947i 0 7.91937 −5.00000 0 9.99806 − 15.5897i 4.52026i 0 − 1.41974i
251.11 2.39619i 0 2.25829 −5.00000 0 −0.977216 − 18.4945i 24.5808i 0 − 11.9809i
251.12 2.56798i 0 1.40549 −5.00000 0 16.0553 − 9.23194i 24.1531i 0 − 12.8399i
251.13 3.85291i 0 −6.84495 −5.00000 0 −11.0192 + 14.8855i 4.45031i 0 − 19.2646i
251.14 4.46361i 0 −11.9238 −5.00000 0 13.0368 + 13.1545i − 17.5145i 0 − 22.3181i
251.15 4.78031i 0 −14.8513 −5.00000 0 18.0094 − 4.32008i − 32.7514i 0 − 23.9015i
251.16 5.09529i 0 −17.9620 −5.00000 0 −10.7370 − 15.0903i − 50.7591i 0 − 25.4764i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c 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.b.a ✓ 16
3.b odd 2 1 315.4.b.b yes 16
7.b odd 2 1 315.4.b.b yes 16
21.c even 2 1 inner 315.4.b.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.4.b.a ✓ 16 1.a even 1 1 trivial
315.4.b.a ✓ 16 21.c even 2 1 inner
315.4.b.b yes 16 3.b odd 2 1
315.4.b.b yes 16 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{8} - 28914 T_{17}^{6} - 437160 T_{17}^{5} + 265069608 T_{17}^{4} + 7628716800 T_{17}^{3} + \cdots - 343459278000000 \) acting on \(S_{4}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 96 T^{14} + \cdots + 576 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T + 5)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 58\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots - 343459278000000)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 28\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 14\!\cdots\!92)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 58\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 34\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 57\!\cdots\!12)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
show more
show less