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Properties

Label 2100.2.s.a
Level $2100$
Weight $2$
Character orbit 2100.s
Analytic conductor $16.769$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,2,Mod(1457,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.1457");
 
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.s (of order \(4\), 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 \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 24x^{12} + 424x^{8} - 159x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{9} q^{3} - \beta_{14} q^{7} + (\beta_{11} + \beta_{7}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{3} - \beta_{14} q^{7} + (\beta_{11} + \beta_{7}) q^{9} + ( - \beta_{2} - \beta_1) q^{11} + (\beta_{9} - \beta_{4} - 2 \beta_{3}) q^{13} + (\beta_{15} + 2 \beta_{13} - \beta_{8}) q^{17} + (2 \beta_{12} - 2 \beta_{11} + 2 \beta_{10}) q^{19} - \beta_1 q^{21} + ( - 2 \beta_{6} + 2 \beta_{4}) q^{23} + ( - 2 \beta_{15} - 2 \beta_{14} + \cdots + \beta_{8}) q^{27}+ \cdots + (5 \beta_{12} + 2 \beta_{11} + \cdots - \beta_{7}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{21} - 4 q^{51} - 16 q^{61} + 92 q^{81} + 40 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 24x^{12} + 424x^{8} - 159x^{4} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -247\nu^{12} - 5984\nu^{8} - 106343\nu^{4} + 16895 ) / 3753 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -83\nu^{12} - 1999\nu^{8} - 35446\nu^{4} + 7585 ) / 1251 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -317\nu^{13} - 7690\nu^{9} - 136192\nu^{5} + 16831\nu ) / 7506 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -107\nu^{13} - 2572\nu^{9} - 45394\nu^{5} + 14521\nu ) / 2502 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 43\nu^{12} + 1044\nu^{8} + 18449\nu^{4} - 3114 ) / 417 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -319\nu^{13} - 7703\nu^{9} - 136187\nu^{5} + 37703\nu ) / 3753 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -313\nu^{14} - 7664\nu^{10} - 136202\nu^{6} - 17407\nu^{2} ) / 7506 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 791\nu^{15} + 19528\nu^{11} + 348928\nu^{7} + 113063\nu^{3} ) / 30024 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -815\nu^{13} - 19684\nu^{9} - 348868\nu^{5} + 69847\nu ) / 7506 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -269\nu^{14} - 6544\nu^{10} - 116296\nu^{6} + 2959\nu^{2} ) / 5004 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -161\nu^{14} - 3896\nu^{10} - 69028\nu^{6} + 12335\nu^{2} ) / 1668 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -1457\nu^{14} - 35116\nu^{10} - 621232\nu^{6} + 164479\nu^{2} ) / 15012 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1433\nu^{15} + 34960\nu^{11} + 621292\nu^{7} + 10925\nu^{3} ) / 15012 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3721\nu^{15} + 89864\nu^{11} + 1591352\nu^{7} - 352847\nu^{3} ) / 30024 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2201\nu^{15} + 53296\nu^{11} + 944632\nu^{7} - 145975\nu^{3} ) / 10008 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{4} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{12} + 3\beta_{11} - \beta_{10} - \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{15} - 5\beta_{14} - 3\beta_{13} + \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{5} + 4\beta_{2} - 15\beta _1 - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -18\beta_{9} - 7\beta_{6} + 29\beta_{4} + 31\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 18\beta_{12} - 11\beta_{11} - 43\beta_{10} + 39\beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -96\beta_{15} + 153\beta_{14} - 7\beta_{13} + 107\beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 171\beta_{5} + 114\beta_{2} + 153\beta _1 - 103 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 438\beta_{9} - 203\beta_{6} - 121\beta_{4} - 595\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 520\beta_{12} - 1065\beta_{11} + 1421\beta_{10} - 577\beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 634\beta_{15} - 1649\beta_{14} + 1503\beta_{13} - 2981\beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( -1129\beta_{5} - 4484\beta_{2} + 2721\beta _1 + 6507 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( -2892\beta_{9} + 7985\beta_{6} - 9577\beta_{4} + 1015\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( -20454\beta_{12} + 30697\beta_{11} - 16027\beta_{10} - 2835\beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 26124\beta_{15} - 26067\beta_{14} - 33589\beta_{13} + 26327\beta_{8} ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(1\) \(\beta_{11}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1457.1
−1.11470 − 1.82181i
−0.0465948 + 0.660512i
0.660512 − 0.0465948i
1.82181 + 1.11470i
−1.82181 − 1.11470i
−0.660512 + 0.0465948i
0.0465948 − 0.660512i
1.11470 + 1.82181i
−1.11470 + 1.82181i
−0.0465948 − 0.660512i
0.660512 + 0.0465948i
1.82181 − 1.11470i
−1.82181 + 1.11470i
−0.660512 − 0.0465948i
0.0465948 + 0.660512i
1.11470 − 1.82181i
0 −1.73015 + 0.0811423i 0 0 0 0.707107 + 0.707107i 0 2.98683 − 0.280776i 0
1457.2 0 −1.64534 − 0.541157i 0 0 0 −0.707107 − 0.707107i 0 2.41430 + 1.78078i 0
1457.3 0 −0.541157 − 1.64534i 0 0 0 0.707107 + 0.707107i 0 −2.41430 + 1.78078i 0
1457.4 0 −0.0811423 + 1.73015i 0 0 0 0.707107 + 0.707107i 0 −2.98683 − 0.280776i 0
1457.5 0 0.0811423 − 1.73015i 0 0 0 −0.707107 − 0.707107i 0 −2.98683 − 0.280776i 0
1457.6 0 0.541157 + 1.64534i 0 0 0 −0.707107 − 0.707107i 0 −2.41430 + 1.78078i 0
1457.7 0 1.64534 + 0.541157i 0 0 0 0.707107 + 0.707107i 0 2.41430 + 1.78078i 0
1457.8 0 1.73015 − 0.0811423i 0 0 0 −0.707107 − 0.707107i 0 2.98683 − 0.280776i 0
1793.1 0 −1.73015 − 0.0811423i 0 0 0 0.707107 − 0.707107i 0 2.98683 + 0.280776i 0
1793.2 0 −1.64534 + 0.541157i 0 0 0 −0.707107 + 0.707107i 0 2.41430 − 1.78078i 0
1793.3 0 −0.541157 + 1.64534i 0 0 0 0.707107 − 0.707107i 0 −2.41430 − 1.78078i 0
1793.4 0 −0.0811423 − 1.73015i 0 0 0 0.707107 − 0.707107i 0 −2.98683 + 0.280776i 0
1793.5 0 0.0811423 + 1.73015i 0 0 0 −0.707107 + 0.707107i 0 −2.98683 + 0.280776i 0
1793.6 0 0.541157 − 1.64534i 0 0 0 −0.707107 + 0.707107i 0 −2.41430 − 1.78078i 0
1793.7 0 1.64534 − 0.541157i 0 0 0 0.707107 − 0.707107i 0 2.41430 − 1.78078i 0
1793.8 0 1.73015 + 0.0811423i 0 0 0 −0.707107 + 0.707107i 0 2.98683 + 0.280776i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1457.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.s.a ✓ 16
3.b odd 2 1 inner 2100.2.s.a ✓ 16
5.b even 2 1 inner 2100.2.s.a ✓ 16
5.c odd 4 2 inner 2100.2.s.a ✓ 16
15.d odd 2 1 inner 2100.2.s.a ✓ 16
15.e even 4 2 inner 2100.2.s.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.s.a ✓ 16 1.a even 1 1 trivial
2100.2.s.a ✓ 16 3.b odd 2 1 inner
2100.2.s.a ✓ 16 5.b even 2 1 inner
2100.2.s.a ✓ 16 5.c odd 4 2 inner
2100.2.s.a ✓ 16 15.d odd 2 1 inner
2100.2.s.a ✓ 16 15.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{4} + 15T_{11}^{2} + 52 \) acting on \(S_{2}^{\mathrm{new}}(2100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 23 T^{12} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 15 T^{2} + 52)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} + 433 T^{4} + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 1817 T^{4} + 692224)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 52 T^{2} + 64)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + 4768 T^{4} + 43264)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 115 T^{2} + 208)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 68)^{8} \) Copy content Toggle raw display
$37$ \( (T^{8} + 784 T^{4} + 65536)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 60 T^{2} + 832)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 12544 T^{4} + 16777216)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 121 T^{4} + 2704)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 4768 T^{4} + 43264)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 236 T^{2} + 13312)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T - 16)^{8} \) Copy content Toggle raw display
$67$ \( (T^{8} + 15376 T^{4} + 1048576)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 200 T^{2} + 208)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 1296)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 33 T^{2} + 64)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 4768 T^{4} + 43264)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 240 T^{2} + 13312)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + 161 T^{4} + 16)^{2} \) Copy content Toggle raw display
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