↑

Properties

Label 2100.2.bi.l
Level $2100$
Weight $2$
Character orbit 2100.bi
Analytic conductor $16.769$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,2,Mod(101,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 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("2100.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.bi (of order \(6\), degree \(2\), 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: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 9 x^{14} - 18 x^{13} + 32 x^{12} - 36 x^{11} + 51 x^{9} - 167 x^{8} + 153 x^{7} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_{10} + \beta_{7} - 1) q^{7} + ( - \beta_{13} - \beta_{10}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (\beta_{10} + \beta_{7} - 1) q^{7} + ( - \beta_{13} - \beta_{10}) q^{9} + (\beta_{15} - \beta_{14} + \cdots + \beta_{2}) q^{11}+ \cdots + (\beta_{15} + \beta_{14} - \beta_{13} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 3 q^{3} - 6 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 3 q^{3} - 6 q^{7} - 9 q^{9} - 18 q^{19} - 11 q^{21} - 18 q^{31} - 12 q^{33} + 6 q^{37} + 12 q^{39} - 4 q^{43} - 18 q^{49} - q^{51} - 6 q^{57} + 36 q^{61} - 19 q^{63} - 30 q^{67} - 54 q^{73} + 7 q^{81} - 81 q^{87} + 20 q^{91} + 34 q^{93} - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 9 x^{14} - 18 x^{13} + 32 x^{12} - 36 x^{11} + 51 x^{9} - 167 x^{8} + 153 x^{7} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{15} + 3 \nu^{14} - 9 \nu^{13} + 18 \nu^{12} - 32 \nu^{11} + 36 \nu^{10} - 51 \nu^{8} + \cdots + 6561 ) / 2187 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1153 \nu^{15} + 6454 \nu^{14} - 29220 \nu^{13} + 76824 \nu^{12} - 121504 \nu^{11} + 151832 \nu^{10} + \cdots + 20670066 ) / 609687 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5395 \nu^{15} - 4206 \nu^{14} + 45069 \nu^{13} - 149400 \nu^{12} + 258388 \nu^{11} + \cdots - 54877662 ) / 1829061 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5968 \nu^{15} + 46258 \nu^{14} - 149151 \nu^{13} + 304524 \nu^{12} - 438368 \nu^{11} + \cdots + 52235766 ) / 1829061 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6954 \nu^{15} - 71512 \nu^{14} + 241188 \nu^{13} - 534429 \nu^{12} + 786063 \nu^{11} + \cdots - 108434376 ) / 1829061 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 23263 \nu^{15} + 951 \nu^{14} - 143820 \nu^{13} + 498699 \nu^{12} - 887410 \nu^{11} + \cdots + 185919057 ) / 5487183 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 24626 \nu^{15} + 32013 \nu^{14} - 242181 \nu^{13} + 767331 \nu^{12} - 1365137 \nu^{11} + \cdots + 276613947 ) / 5487183 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 9427 \nu^{15} + 56618 \nu^{14} - 170922 \nu^{13} + 329391 \nu^{12} - 457733 \nu^{11} + \cdots + 47911338 ) / 1829061 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 28354 \nu^{15} + 95439 \nu^{14} - 197100 \nu^{13} + 247392 \nu^{12} - 215912 \nu^{11} + \cdots - 33668865 ) / 5487183 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11554 \nu^{15} + 65950 \nu^{14} - 194877 \nu^{13} + 389808 \nu^{12} - 532664 \nu^{11} + \cdots + 60732990 ) / 1829061 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 35773 \nu^{15} - 50757 \nu^{14} - 13698 \nu^{13} + 367722 \nu^{12} - 769078 \nu^{11} + \cdots + 220725162 ) / 5487183 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 35881 \nu^{15} - 110493 \nu^{14} + 242262 \nu^{13} - 315135 \nu^{12} + 321290 \nu^{11} + \cdots + 33668865 ) / 5487183 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 21897 \nu^{15} + 62870 \nu^{14} - 114183 \nu^{13} + 108063 \nu^{12} - 49284 \nu^{11} + \cdots - 49687182 ) / 1829061 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 8452 \nu^{15} + 21901 \nu^{14} - 40300 \nu^{13} + 29256 \nu^{12} - 302 \nu^{11} + \cdots - 23491053 ) / 609687 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} + \beta_{10} + \beta_{9} + \beta_{6} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{14} - \beta_{7} - \beta_{5} + 2\beta_{3} + 2\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{14} + 2\beta_{10} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + 3\beta_{4} - \beta_{3} - 2\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{15} + \beta_{14} - 2 \beta_{13} - \beta_{12} + 2 \beta_{11} - 3 \beta_{10} - \beta_{9} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{15} + 2 \beta_{14} + 4 \beta_{13} + 4 \beta_{12} + \beta_{11} - 2 \beta_{9} - \beta_{8} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 5 \beta_{15} + 5 \beta_{14} - 2 \beta_{13} - 4 \beta_{12} + 5 \beta_{11} - 19 \beta_{10} - 7 \beta_{9} + \cdots + 38 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3 \beta_{15} - \beta_{14} + 12 \beta_{12} + 2 \beta_{11} + 32 \beta_{10} - 3 \beta_{9} - \beta_{8} + \cdots - 32 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 14 \beta_{15} - 14 \beta_{14} - 29 \beta_{13} + 29 \beta_{12} - 13 \beta_{11} + 30 \beta_{10} + \cdots - 15 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 26 \beta_{15} - 37 \beta_{14} + 28 \beta_{13} - 26 \beta_{11} + 50 \beta_{10} + 39 \beta_{9} + \cdots - 66 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 50 \beta_{15} + 9 \beta_{14} + 40 \beta_{13} + 20 \beta_{12} + 59 \beta_{11} + 80 \beta_{10} + \cdots + 80 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 12 \beta_{15} + 12 \beta_{14} + 105 \beta_{13} + 105 \beta_{12} + 110 \beta_{11} - 72 \beta_{9} + \cdots - 239 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 102 \beta_{15} + 162 \beta_{14} + 120 \beta_{13} + 240 \beta_{12} + 102 \beta_{11} - 192 \beta_{10} + \cdots + 384 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 174 \beta_{15} + 174 \beta_{14} - 388 \beta_{12} + 348 \beta_{11} - 91 \beta_{10} - 694 \beta_{9} + \cdots + 91 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 140 \beta_{15} - 140 \beta_{14} - 408 \beta_{13} + 408 \beta_{12} - 138 \beta_{11} + 2208 \beta_{10} + \cdots - 1104 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(\beta_{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
1.72685 + 0.134063i
1.25639 + 1.19226i
0.990987 + 1.42054i
0.747325 − 1.56253i
−0.404332 − 1.68420i
−0.417865 + 1.68089i
−0.734734 − 1.56849i
−1.66463 − 0.478563i
1.72685 − 0.134063i
1.25639 − 1.19226i
0.990987 − 1.42054i
0.747325 + 1.56253i
−0.404332 + 1.68420i
−0.417865 − 1.68089i
−0.734734 + 1.56849i
−1.66463 + 0.478563i
0 −1.72685 + 0.134063i 0 0 0 0.786875 + 2.52603i 0 2.96405 − 0.463016i 0
101.2 0 −1.25639 + 1.19226i 0 0 0 1.60761 − 2.10133i 0 0.157032 − 2.99589i 0
101.3 0 −0.990987 + 1.42054i 0 0 0 −2.64111 − 0.156656i 0 −1.03589 − 2.81548i 0
101.4 0 −0.747325 − 1.56253i 0 0 0 0.786875 + 2.52603i 0 −1.88301 + 2.33544i 0
101.5 0 0.404332 − 1.68420i 0 0 0 1.60761 − 2.10133i 0 −2.67303 − 1.36195i 0
101.6 0 0.417865 + 1.68089i 0 0 0 −1.25338 + 2.33003i 0 −2.65078 + 1.40477i 0
101.7 0 0.734734 − 1.56849i 0 0 0 −2.64111 − 0.156656i 0 −1.92033 − 2.30485i 0
101.8 0 1.66463 − 0.478563i 0 0 0 −1.25338 + 2.33003i 0 2.54196 − 1.59326i 0
1601.1 0 −1.72685 − 0.134063i 0 0 0 0.786875 − 2.52603i 0 2.96405 + 0.463016i 0
1601.2 0 −1.25639 − 1.19226i 0 0 0 1.60761 + 2.10133i 0 0.157032 + 2.99589i 0
1601.3 0 −0.990987 − 1.42054i 0 0 0 −2.64111 + 0.156656i 0 −1.03589 + 2.81548i 0
1601.4 0 −0.747325 + 1.56253i 0 0 0 0.786875 − 2.52603i 0 −1.88301 − 2.33544i 0
1601.5 0 0.404332 + 1.68420i 0 0 0 1.60761 + 2.10133i 0 −2.67303 + 1.36195i 0
1601.6 0 0.417865 − 1.68089i 0 0 0 −1.25338 − 2.33003i 0 −2.65078 − 1.40477i 0
1601.7 0 0.734734 + 1.56849i 0 0 0 −2.64111 + 0.156656i 0 −1.92033 + 2.30485i 0
1601.8 0 1.66463 + 0.478563i 0 0 0 −1.25338 − 2.33003i 0 2.54196 + 1.59326i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.bi.l ✓ 16
3.b odd 2 1 inner 2100.2.bi.l ✓ 16
5.b even 2 1 2100.2.bi.m yes 16
5.c odd 4 2 2100.2.bo.i 32
7.d odd 6 1 inner 2100.2.bi.l ✓ 16
15.d odd 2 1 2100.2.bi.m yes 16
15.e even 4 2 2100.2.bo.i 32
21.g even 6 1 inner 2100.2.bi.l ✓ 16
35.i odd 6 1 2100.2.bi.m yes 16
35.k even 12 2 2100.2.bo.i 32
105.p even 6 1 2100.2.bi.m yes 16
105.w odd 12 2 2100.2.bo.i 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.bi.l ✓ 16 1.a even 1 1 trivial
2100.2.bi.l ✓ 16 3.b odd 2 1 inner
2100.2.bi.l ✓ 16 7.d odd 6 1 inner
2100.2.bi.l ✓ 16 21.g even 6 1 inner
2100.2.bi.m yes 16 5.b even 2 1
2100.2.bi.m yes 16 15.d odd 2 1
2100.2.bi.m yes 16 35.i odd 6 1
2100.2.bi.m yes 16 105.p even 6 1
2100.2.bo.i 32 5.c odd 4 2
2100.2.bo.i 32 15.e even 4 2
2100.2.bo.i 32 35.k even 12 2
2100.2.bo.i 32 105.w odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2100, [\chi])\):

\( T_{11}^{16} - 74 T_{11}^{14} + 3715 T_{11}^{12} - 101180 T_{11}^{10} + 1984747 T_{11}^{8} + \cdots + 1475789056 \) Copy content Toggle raw display
\( T_{13}^{8} + 21T_{13}^{6} + 71T_{13}^{4} + 78T_{13}^{2} + 25 \) Copy content Toggle raw display
\( T_{19}^{8} + 9T_{19}^{7} - 12T_{19}^{6} - 351T_{19}^{5} + 596T_{19}^{4} + 13104T_{19}^{3} + 40869T_{19}^{2} + 27888T_{19} + 6889 \) Copy content Toggle raw display
\( T_{37}^{8} - 3 T_{37}^{7} + 148 T_{37}^{6} - 315 T_{37}^{5} + 16836 T_{37}^{4} - 29376 T_{37}^{3} + \cdots + 12837889 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 3 T^{15} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + 3 T^{7} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 1475789056 \) Copy content Toggle raw display
$13$ \( (T^{8} + 21 T^{6} + \cdots + 25)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 122 T^{14} + \cdots + 614656 \) Copy content Toggle raw display
$19$ \( (T^{8} + 9 T^{7} + \cdots + 6889)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 157351936 \) Copy content Toggle raw display
$29$ \( (T^{8} + 142 T^{6} + \cdots + 78400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 9 T^{7} + \cdots + 148996)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 3 T^{7} + \cdots + 12837889)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 153 T^{6} + \cdots + 784)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + T^{3} - 22 T^{2} + \cdots + 88)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 123259377746176 \) Copy content Toggle raw display
$53$ \( T^{16} - 162 T^{14} + \cdots + 614656 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 51336683776 \) Copy content Toggle raw display
$61$ \( (T^{8} - 18 T^{7} + \cdots + 477481)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 15 T^{7} + \cdots + 129600)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 429 T^{6} + \cdots + 3013696)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 27 T^{7} + \cdots + 1577536)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 121 T^{6} + \cdots + 2217121)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 258 T^{6} + \cdots + 283024)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 58\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{8} + 374 T^{6} + \cdots + 26388769)^{2} \) Copy content Toggle raw display
show more
show less