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Properties

Label 2940.2.d.c
Level $2940$
Weight $2$
Character orbit 2940.d
Analytic conductor $23.476$
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] = [2940,2,Mod(881,2940)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2940, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("2940.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2940.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: \(23.4760181943\)
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: \( x^{16} - 4 x^{15} + 10 x^{14} - 16 x^{13} + 7 x^{12} + 8 x^{11} - 14 x^{10} - 28 x^{9} + 152 x^{8} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + q^{5} + (\beta_{8} + \beta_{6} - \beta_{4} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + q^{5} + (\beta_{8} + \beta_{6} - \beta_{4} + \cdots + \beta_1) q^{9}+ \cdots + (\beta_{13} - 2 \beta_{12} - 3 \beta_{10} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{3} + 16 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{3} + 16 q^{5} - 4 q^{9} - 4 q^{15} + 24 q^{17} + 16 q^{25} + 8 q^{27} - 12 q^{33} + 12 q^{39} - 16 q^{41} + 16 q^{43} - 4 q^{45} + 40 q^{47} - 20 q^{51} - 16 q^{57} + 16 q^{59} + 16 q^{67} - 4 q^{75} + 24 q^{79} + 44 q^{81} - 48 q^{83} + 24 q^{85} - 12 q^{87} + 32 q^{89} - 16 q^{93} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 10 x^{14} - 16 x^{13} + 7 x^{12} + 8 x^{11} - 14 x^{10} - 28 x^{9} + 152 x^{8} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11 \nu^{15} + 25 \nu^{14} - 157 \nu^{13} + 289 \nu^{12} - 586 \nu^{11} - 86 \nu^{10} + 1136 \nu^{9} + \cdots + 312741 ) / 69984 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{15} + 4 \nu^{14} - 10 \nu^{13} + 16 \nu^{12} - 7 \nu^{11} - 8 \nu^{10} + 14 \nu^{9} + \cdots + 8748 ) / 2187 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 23 \nu^{15} - 89 \nu^{14} + 155 \nu^{13} - 221 \nu^{12} - 58 \nu^{11} + 430 \nu^{10} + 368 \nu^{9} + \cdots - 47385 ) / 23328 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 32 \nu^{15} + 389 \nu^{14} - 1112 \nu^{13} + 1439 \nu^{12} - 1124 \nu^{11} - 2650 \nu^{10} + \cdots + 885735 ) / 34992 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13 \nu^{15} + 49 \nu^{14} - 117 \nu^{13} + 153 \nu^{12} + 6 \nu^{11} - 198 \nu^{10} + 240 \nu^{9} + \cdots + 81405 ) / 7776 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10 \nu^{15} + 51 \nu^{14} - 138 \nu^{13} + 156 \nu^{12} + 12 \nu^{11} - 270 \nu^{10} + \cdots + 72900 ) / 5832 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 143 \nu^{15} - 605 \nu^{14} + 1355 \nu^{13} - 1817 \nu^{12} + 134 \nu^{11} + 2902 \nu^{10} + \cdots - 907605 ) / 69984 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 29 \nu^{15} + 88 \nu^{14} - 103 \nu^{13} + 100 \nu^{12} + 266 \nu^{11} - 332 \nu^{10} + \cdots - 23328 ) / 11664 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 29 \nu^{15} + 158 \nu^{14} - 293 \nu^{13} + 332 \nu^{12} + 46 \nu^{11} - 796 \nu^{10} + \cdots + 137052 ) / 11664 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 28 \nu^{15} - 187 \nu^{14} + 364 \nu^{13} - 469 \nu^{12} + 100 \nu^{11} + 1238 \nu^{10} + \cdots - 190269 ) / 11664 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 14 \nu^{15} + \nu^{14} + 44 \nu^{13} - 83 \nu^{12} + 188 \nu^{11} + 178 \nu^{10} - 460 \nu^{9} + \cdots - 63423 ) / 3888 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 85 \nu^{15} - 415 \nu^{14} + 901 \nu^{13} - 1087 \nu^{12} + 34 \nu^{11} + 2114 \nu^{10} + \cdots - 474579 ) / 23328 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 188 \nu^{15} - 491 \nu^{14} + 458 \nu^{13} - 371 \nu^{12} - 1672 \nu^{11} + 1090 \nu^{10} + \cdots + 129033 ) / 34992 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 220 \nu^{15} - 901 \nu^{14} + 1996 \nu^{13} - 1903 \nu^{12} - 140 \nu^{11} + 3818 \nu^{10} + \cdots - 1025703 ) / 34992 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{9} + \beta_{8} + \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{14} - \beta_{11} - \beta_{10} - 2\beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + 3\beta_{3} - 2\beta_{2} - \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} + 2 \beta_{14} - \beta_{13} + 2 \beta_{12} + 2 \beta_{11} + 4 \beta_{10} + \beta_{8} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{15} - 5 \beta_{14} + 4 \beta_{13} - 5 \beta_{12} + \beta_{11} - 6 \beta_{10} - \beta_{8} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3 \beta_{15} + 2 \beta_{14} + 7 \beta_{13} + 6 \beta_{12} + \beta_{11} + 8 \beta_{10} + 2 \beta_{9} + \cdots - 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4 \beta_{15} - 10 \beta_{14} + 4 \beta_{13} - \beta_{12} + 6 \beta_{11} - 11 \beta_{10} + 14 \beta_{9} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 10 \beta_{15} - 8 \beta_{14} - 5 \beta_{13} + 16 \beta_{12} - 3 \beta_{11} + 24 \beta_{10} - 57 \beta_{9} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 12 \beta_{14} - 12 \beta_{13} - 18 \beta_{12} + 60 \beta_{11} + 42 \beta_{10} - 12 \beta_{9} + \cdots - 48 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 48 \beta_{14} + 62 \beta_{13} - 60 \beta_{12} + 54 \beta_{11} - 12 \beta_{10} + 38 \beta_{9} + \cdots + 108 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 108 \beta_{15} - 8 \beta_{14} - 8 \beta_{13} + 60 \beta_{12} + 64 \beta_{11} + 28 \beta_{10} + \cdots - 279 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 164 \beta_{15} - 108 \beta_{14} - 110 \beta_{13} + 88 \beta_{12} + 60 \beta_{11} + 76 \beta_{10} + \cdots + 84 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 24 \beta_{15} - 254 \beta_{14} + 100 \beta_{13} - 128 \beta_{12} + 70 \beta_{11} + 538 \beta_{10} + \cdots - 828 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 62 \beta_{15} - 280 \beta_{14} - 345 \beta_{13} - 752 \beta_{12} + 372 \beta_{11} + 616 \beta_{10} + \cdots + 612 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(1177\) \(1471\) \(1961\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
1.71731 + 0.225499i
1.71731 − 0.225499i
1.65131 + 0.522651i
1.65131 − 0.522651i
0.683652 + 1.59142i
0.683652 − 1.59142i
0.595231 + 1.62656i
0.595231 − 1.62656i
0.354244 + 1.69544i
0.354244 − 1.69544i
−0.0726671 + 1.73053i
−0.0726671 − 1.73053i
−1.26230 + 1.18600i
−1.26230 − 1.18600i
−1.66678 + 0.470990i
−1.66678 − 0.470990i
0 −1.71731 − 0.225499i 0 1.00000 0 0 0 2.89830 + 0.774504i 0
881.2 0 −1.71731 + 0.225499i 0 1.00000 0 0 0 2.89830 − 0.774504i 0
881.3 0 −1.65131 − 0.522651i 0 1.00000 0 0 0 2.45367 + 1.72612i 0
881.4 0 −1.65131 + 0.522651i 0 1.00000 0 0 0 2.45367 − 1.72612i 0
881.5 0 −0.683652 − 1.59142i 0 1.00000 0 0 0 −2.06524 + 2.17596i 0
881.6 0 −0.683652 + 1.59142i 0 1.00000 0 0 0 −2.06524 − 2.17596i 0
881.7 0 −0.595231 − 1.62656i 0 1.00000 0 0 0 −2.29140 + 1.93636i 0
881.8 0 −0.595231 + 1.62656i 0 1.00000 0 0 0 −2.29140 − 1.93636i 0
881.9 0 −0.354244 − 1.69544i 0 1.00000 0 0 0 −2.74902 + 1.20120i 0
881.10 0 −0.354244 + 1.69544i 0 1.00000 0 0 0 −2.74902 − 1.20120i 0
881.11 0 0.0726671 − 1.73053i 0 1.00000 0 0 0 −2.98944 − 0.251505i 0
881.12 0 0.0726671 + 1.73053i 0 1.00000 0 0 0 −2.98944 + 0.251505i 0
881.13 0 1.26230 − 1.18600i 0 1.00000 0 0 0 0.186794 − 2.99418i 0
881.14 0 1.26230 + 1.18600i 0 1.00000 0 0 0 0.186794 + 2.99418i 0
881.15 0 1.66678 − 0.470990i 0 1.00000 0 0 0 2.55634 − 1.57008i 0
881.16 0 1.66678 + 0.470990i 0 1.00000 0 0 0 2.55634 + 1.57008i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.16
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 2940.2.d.c ✓ 16
3.b odd 2 1 2940.2.d.d yes 16
7.b odd 2 1 2940.2.d.d yes 16
21.c even 2 1 inner 2940.2.d.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2940.2.d.c ✓ 16 1.a even 1 1 trivial
2940.2.d.c ✓ 16 21.c even 2 1 inner
2940.2.d.d yes 16 3.b odd 2 1
2940.2.d.d yes 16 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2940, [\chi])\):

\( T_{11}^{16} + 92 T_{11}^{14} + 3190 T_{11}^{12} + 52268 T_{11}^{10} + 416433 T_{11}^{8} + 1425264 T_{11}^{6} + \cdots + 1024 \) Copy content Toggle raw display
\( T_{17}^{8} - 12T_{17}^{7} + 6T_{17}^{6} + 316T_{17}^{5} - 605T_{17}^{4} - 1888T_{17}^{3} + 3948T_{17}^{2} + 304T_{17} - 356 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 4 T^{15} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( (T - 1)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + 92 T^{14} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( T^{16} + 124 T^{14} + \cdots + 295936 \) Copy content Toggle raw display
$17$ \( (T^{8} - 12 T^{7} + \cdots - 356)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + 168 T^{14} + \cdots + 47554816 \) Copy content Toggle raw display
$23$ \( T^{16} + 152 T^{14} + \cdots + 226576 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 951599104 \) Copy content Toggle raw display
$31$ \( T^{16} + 160 T^{14} + \cdots + 1317904 \) Copy content Toggle raw display
$37$ \( (T^{8} - 200 T^{6} + \cdots - 152096)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 8 T^{7} + \cdots - 4384)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 8 T^{7} + \cdots + 512)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 20 T^{7} + \cdots + 7345084)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 28125973264 \) Copy content Toggle raw display
$59$ \( (T^{8} - 8 T^{7} + \cdots - 61312)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 34815097786624 \) Copy content Toggle raw display
$67$ \( (T^{8} - 8 T^{7} + \cdots + 34259072)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 972164448256 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 2740731904 \) Copy content Toggle raw display
$79$ \( (T^{8} - 12 T^{7} + \cdots - 1472476)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 24 T^{7} + \cdots - 369664)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 16 T^{7} + \cdots + 7974400)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
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