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Properties

Label 420.2.c.a
Level $420$
Weight $2$
Character orbit 420.c
Analytic conductor $3.354$
Analytic rank $0$
Dimension $16$
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] = [420,2,Mod(391,420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(420, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("420.391");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.c (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: \(3.35371688489\)
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: \( x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 3 x^{12} + 2 x^{11} - 7 x^{10} + 12 x^{9} - 28 x^{8} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + \beta_{2} q^{4} + \beta_{4} q^{5} - \beta_1 q^{6} + (\beta_{15} + \beta_{10}) q^{7} + ( - \beta_{7} + \beta_{5} + \cdots - \beta_1) q^{8}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + \beta_{2} q^{4} + \beta_{4} q^{5} - \beta_1 q^{6} + (\beta_{15} + \beta_{10}) q^{7} + ( - \beta_{7} + \beta_{5} + \cdots - \beta_1) q^{8}+ \cdots + (\beta_{15} - \beta_{14} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{2} - 16 q^{3} - 2 q^{4} - 2 q^{6} - 4 q^{7} + 2 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{2} - 16 q^{3} - 2 q^{4} - 2 q^{6} - 4 q^{7} + 2 q^{8} + 16 q^{9} + 2 q^{12} - 2 q^{14} + 6 q^{16} + 2 q^{18} - 24 q^{19} + 4 q^{21} - 12 q^{22} - 2 q^{24} - 16 q^{25} - 12 q^{26} - 16 q^{27} + 14 q^{28} + 16 q^{29} + 8 q^{31} - 18 q^{32} + 24 q^{34} - 2 q^{36} + 24 q^{37} + 28 q^{38} + 12 q^{40} + 2 q^{42} - 8 q^{44} - 20 q^{46} + 16 q^{47} - 6 q^{48} - 16 q^{49} - 2 q^{50} - 20 q^{52} - 32 q^{53} - 2 q^{54} + 2 q^{56} + 24 q^{57} - 32 q^{58} + 8 q^{59} + 16 q^{62} - 4 q^{63} - 2 q^{64} - 8 q^{65} + 12 q^{66} + 4 q^{68} + 2 q^{72} - 4 q^{74} + 16 q^{75} + 16 q^{76} - 8 q^{77} + 12 q^{78} - 16 q^{80} + 16 q^{81} - 4 q^{82} + 8 q^{83} - 14 q^{84} + 64 q^{86} - 16 q^{87} - 52 q^{88} + 16 q^{91} + 64 q^{92} - 8 q^{93} + 16 q^{94} + 18 q^{96} - 86 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 3 x^{12} + 2 x^{11} - 7 x^{10} + 12 x^{9} - 28 x^{8} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{15} + 4 \nu^{14} + 5 \nu^{13} - 6 \nu^{12} - 7 \nu^{11} - 20 \nu^{10} - 33 \nu^{9} + \cdots + 384 ) / 512 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3 \nu^{14} - 5 \nu^{12} + 2 \nu^{11} + 7 \nu^{10} + 17 \nu^{8} - 18 \nu^{7} + 20 \nu^{6} + \cdots - 384 ) / 128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5 \nu^{15} + 4 \nu^{14} - 15 \nu^{13} + 18 \nu^{12} + 5 \nu^{11} - 4 \nu^{10} + 3 \nu^{9} + \cdots + 640 ) / 512 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{15} + 5 \nu^{14} + \nu^{13} + 5 \nu^{12} - 13 \nu^{11} - 21 \nu^{10} - 5 \nu^{9} - 9 \nu^{8} + \cdots + 576 ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5 \nu^{15} + 16 \nu^{14} - 15 \nu^{13} + 38 \nu^{12} - 3 \nu^{11} - 32 \nu^{10} + 3 \nu^{9} + \cdots + 2176 ) / 512 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5 \nu^{15} - 28 \nu^{14} + \nu^{13} - 14 \nu^{12} + 53 \nu^{11} + 60 \nu^{10} + 51 \nu^{9} + \cdots - 2944 ) / 512 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{15} - 2 \nu^{14} + 3 \nu^{13} - 4 \nu^{12} + 3 \nu^{11} + 2 \nu^{10} - 7 \nu^{9} + 12 \nu^{8} + \cdots - 256 ) / 64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3 \nu^{15} + 5 \nu^{13} - 2 \nu^{12} - 7 \nu^{11} - 17 \nu^{9} + 18 \nu^{8} - 20 \nu^{7} + \cdots + 384 \nu ) / 128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11 \nu^{15} - 8 \nu^{14} - 9 \nu^{13} + 18 \nu^{12} + 43 \nu^{11} + 8 \nu^{10} + 21 \nu^{9} + \cdots + 128 ) / 512 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3 \nu^{15} - 2 \nu^{14} + 3 \nu^{13} - 8 \nu^{12} - \nu^{11} + 14 \nu^{10} + \nu^{9} + 32 \nu^{8} + \cdots - 576 ) / 128 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3 \nu^{15} + 2 \nu^{14} - 5 \nu^{13} + 8 \nu^{12} + 3 \nu^{11} - 2 \nu^{10} + 9 \nu^{9} - 32 \nu^{8} + \cdots + 384 ) / 64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3 \nu^{15} + 6 \nu^{14} - 3 \nu^{13} + 12 \nu^{12} + \nu^{11} - 10 \nu^{10} + 7 \nu^{9} - 36 \nu^{8} + \cdots + 704 ) / 64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 33 \nu^{15} + 40 \nu^{14} - 43 \nu^{13} + 86 \nu^{12} + 17 \nu^{11} - 104 \nu^{10} + 79 \nu^{9} + \cdots + 5504 ) / 512 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{5} - \beta_{4} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} + \beta_{12} + \beta_{9} + \beta_{6} + \beta_{4} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} + \beta_{13} + \beta_{12} + 2\beta_{11} - \beta_{10} - \beta_{8} + \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{15} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - 2\beta_{4} + 3\beta_{3} + \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4 \beta_{15} - 2 \beta_{14} - 4 \beta_{13} + 2 \beta_{12} + 2 \beta_{11} + 4 \beta_{10} - 4 \beta_{9} + \cdots + 3 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2 \beta_{14} - 5 \beta_{13} + \beta_{12} + 2 \beta_{11} - 3 \beta_{9} + 2 \beta_{8} + 4 \beta_{7} + \cdots + 7 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3 \beta_{15} + 2 \beta_{14} - \beta_{13} - 3 \beta_{12} - 4 \beta_{11} - 7 \beta_{10} - 2 \beta_{9} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( \beta_{15} + 2 \beta_{14} + 2 \beta_{13} - 6 \beta_{11} + 9 \beta_{10} - 3 \beta_{9} + \beta_{8} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 10 \beta_{15} - 4 \beta_{14} - 14 \beta_{13} + 10 \beta_{12} + 8 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 14 \beta_{15} + 8 \beta_{14} + 17 \beta_{13} + \beta_{12} + 2 \beta_{10} + 11 \beta_{9} + 6 \beta_{8} + \cdots - 15 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - \beta_{15} + 12 \beta_{14} + 9 \beta_{13} - 3 \beta_{12} - 2 \beta_{11} + 15 \beta_{10} + 8 \beta_{9} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - \beta_{15} + 12 \beta_{14} - 12 \beta_{13} + 24 \beta_{12} - 4 \beta_{11} - 17 \beta_{10} - \beta_{9} + \cdots - 20 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 38 \beta_{14} + 8 \beta_{13} + 66 \beta_{12} + 46 \beta_{11} - 24 \beta_{10} + 2 \beta_{8} + \cdots + 12 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/420\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(241\) \(281\) \(337\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
391.1
−1.39396 − 0.238466i
−1.39396 + 0.238466i
−0.947441 − 1.04993i
−0.947441 + 1.04993i
−0.449546 − 1.34086i
−0.449546 + 1.34086i
−0.102186 − 1.41052i
−0.102186 + 1.41052i
0.309204 − 1.38000i
0.309204 + 1.38000i
1.07312 − 0.921096i
1.07312 + 0.921096i
1.10145 − 0.887017i
1.10145 + 0.887017i
1.40936 − 0.117062i
1.40936 + 0.117062i
−1.39396 − 0.238466i −1.00000 1.88627 + 0.664826i 1.00000i 1.39396 + 0.238466i 2.37694 + 1.16196i −2.47085 − 1.37655i 1.00000 0.238466 − 1.39396i
391.2 −1.39396 + 0.238466i −1.00000 1.88627 − 0.664826i − 1.00000i 1.39396 − 0.238466i 2.37694 − 1.16196i −2.47085 + 1.37655i 1.00000 0.238466 + 1.39396i
391.3 −0.947441 − 1.04993i −1.00000 −0.204711 + 1.98950i 1.00000i 0.947441 + 1.04993i −2.29670 + 1.31346i 2.28279 − 1.67000i 1.00000 1.04993 − 0.947441i
391.4 −0.947441 + 1.04993i −1.00000 −0.204711 − 1.98950i − 1.00000i 0.947441 − 1.04993i −2.29670 − 1.31346i 2.28279 + 1.67000i 1.00000 1.04993 + 0.947441i
391.5 −0.449546 − 1.34086i −1.00000 −1.59582 + 1.20556i − 1.00000i 0.449546 + 1.34086i 1.40015 − 2.24490i 2.33388 + 1.59781i 1.00000 −1.34086 + 0.449546i
391.6 −0.449546 + 1.34086i −1.00000 −1.59582 − 1.20556i 1.00000i 0.449546 − 1.34086i 1.40015 + 2.24490i 2.33388 − 1.59781i 1.00000 −1.34086 − 0.449546i
391.7 −0.102186 − 1.41052i −1.00000 −1.97912 + 0.288270i − 1.00000i 0.102186 + 1.41052i −0.178143 + 2.63975i 0.608847 + 2.76212i 1.00000 −1.41052 + 0.102186i
391.8 −0.102186 + 1.41052i −1.00000 −1.97912 − 0.288270i 1.00000i 0.102186 − 1.41052i −0.178143 − 2.63975i 0.608847 − 2.76212i 1.00000 −1.41052 − 0.102186i
391.9 0.309204 − 1.38000i −1.00000 −1.80879 − 0.853401i 1.00000i −0.309204 + 1.38000i −2.64459 − 0.0785232i −1.73698 + 2.23224i 1.00000 1.38000 + 0.309204i
391.10 0.309204 + 1.38000i −1.00000 −1.80879 + 0.853401i − 1.00000i −0.309204 − 1.38000i −2.64459 + 0.0785232i −1.73698 − 2.23224i 1.00000 1.38000 − 0.309204i
391.11 1.07312 − 0.921096i −1.00000 0.303166 − 1.97689i − 1.00000i −1.07312 + 0.921096i −1.82575 + 1.91485i −1.49557 − 2.40068i 1.00000 −0.921096 − 1.07312i
391.12 1.07312 + 0.921096i −1.00000 0.303166 + 1.97689i 1.00000i −1.07312 − 0.921096i −1.82575 − 1.91485i −1.49557 + 2.40068i 1.00000 −0.921096 + 1.07312i
391.13 1.10145 − 0.887017i −1.00000 0.426402 − 1.95402i 1.00000i −1.10145 + 0.887017i 0.391948 − 2.61656i −1.26358 − 2.53049i 1.00000 0.887017 + 1.10145i
391.14 1.10145 + 0.887017i −1.00000 0.426402 + 1.95402i − 1.00000i −1.10145 − 0.887017i 0.391948 + 2.61656i −1.26358 + 2.53049i 1.00000 0.887017 − 1.10145i
391.15 1.40936 − 0.117062i −1.00000 1.97259 − 0.329965i 1.00000i −1.40936 + 0.117062i 0.776136 + 2.52935i 2.74147 − 0.695955i 1.00000 0.117062 + 1.40936i
391.16 1.40936 + 0.117062i −1.00000 1.97259 + 0.329965i − 1.00000i −1.40936 − 0.117062i 0.776136 − 2.52935i 2.74147 + 0.695955i 1.00000 0.117062 − 1.40936i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 391.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 420.2.c.a ✓ 16
3.b odd 2 1 1260.2.c.d 16
4.b odd 2 1 420.2.c.b yes 16
7.b odd 2 1 420.2.c.b yes 16
12.b even 2 1 1260.2.c.e 16
21.c even 2 1 1260.2.c.e 16
28.d even 2 1 inner 420.2.c.a ✓ 16
84.h odd 2 1 1260.2.c.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.c.a ✓ 16 1.a even 1 1 trivial
420.2.c.a ✓ 16 28.d even 2 1 inner
420.2.c.b yes 16 4.b odd 2 1
420.2.c.b yes 16 7.b odd 2 1
1260.2.c.d 16 3.b odd 2 1
1260.2.c.d 16 84.h odd 2 1
1260.2.c.e 16 12.b even 2 1
1260.2.c.e 16 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{19}^{8} + 12T_{19}^{7} + 6T_{19}^{6} - 288T_{19}^{5} - 352T_{19}^{4} + 2400T_{19}^{3} + 1696T_{19}^{2} - 6784T_{19} + 1408 \) acting on \(S_{2}^{\mathrm{new}}(420, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 2 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T + 1)^{16} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + 4 T^{15} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} + 108 T^{14} + \cdots + 16384 \) Copy content Toggle raw display
$13$ \( T^{16} + 108 T^{14} + \cdots + 1982464 \) Copy content Toggle raw display
$17$ \( T^{16} + 104 T^{14} + \cdots + 589824 \) Copy content Toggle raw display
$19$ \( (T^{8} + 12 T^{7} + \cdots + 1408)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 110166016 \) Copy content Toggle raw display
$29$ \( (T^{8} - 8 T^{7} + \cdots - 475392)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 4 T^{7} + \cdots + 435072)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 12 T^{7} + \cdots - 843008)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 186292371456 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 32480690176 \) Copy content Toggle raw display
$47$ \( (T^{8} - 8 T^{7} + \cdots + 1243904)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 16 T^{7} + \cdots + 311424)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 4 T^{7} + \cdots - 323072)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 334139490304 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 1665379926016 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 57232008953856 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 34021064704 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 57415827456 \) Copy content Toggle raw display
$83$ \( (T^{8} - 4 T^{7} + \cdots - 30015488)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 278378643456 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 970650566656 \) Copy content Toggle raw display
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