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Properties

Label 4100.2.g.e
Level $4100$
Weight $2$
Character orbit 4100.g
Analytic conductor $32.739$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4100,2,Mod(2049,4100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4100.2049");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4100.g (of order \(2\), degree \(1\), not 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: \(32.7386648287\)
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: \( x^{16} - 8 x^{15} + 32 x^{14} - 52 x^{13} + 78 x^{12} - 316 x^{11} + 1384 x^{10} - 2216 x^{9} + 1603 x^{8} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 820)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} - \beta_{7} q^{7} + ( - \beta_{6} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} - \beta_{7} q^{7} + ( - \beta_{6} + 1) q^{9} - \beta_{10} q^{11} - \beta_1 q^{13} + (\beta_{7} - \beta_{4} + \beta_1) q^{17} + (\beta_{10} + \beta_{8}) q^{19} + ( - \beta_{14} - \beta_{13} - \beta_{6} + \cdots + 1) q^{21}+ \cdots + ( - 3 \beta_{11} - \beta_{8}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{9} + 12 q^{21} + 28 q^{31} + 20 q^{39} - 12 q^{41} + 36 q^{49} + 32 q^{51} - 20 q^{59} + 16 q^{61} - 40 q^{81} + 12 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 32 x^{14} - 52 x^{13} + 78 x^{12} - 316 x^{11} + 1384 x^{10} - 2216 x^{9} + 1603 x^{8} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 11\!\cdots\!52 \nu^{15} + \cdots + 62\!\cdots\!51 ) / 12\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 90\!\cdots\!62 \nu^{15} + \cdots + 83\!\cdots\!40 ) / 58\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 15\!\cdots\!93 \nu^{15} + \cdots - 35\!\cdots\!74 ) / 72\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25\!\cdots\!84 \nu^{15} + \cdots + 46\!\cdots\!85 ) / 11\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 27\!\cdots\!88 \nu^{15} + \cdots - 25\!\cdots\!29 ) / 11\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 45\!\cdots\!53 \nu^{15} + \cdots + 27\!\cdots\!23 ) / 72\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 26\!\cdots\!40 \nu^{15} + \cdots - 89\!\cdots\!27 ) / 37\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 84\!\cdots\!40 \nu^{15} + \cdots + 74\!\cdots\!23 ) / 41\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 15\!\cdots\!46 \nu^{15} + \cdots - 11\!\cdots\!02 ) / 73\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 99\!\cdots\!58 \nu^{15} + \cdots + 77\!\cdots\!47 ) / 41\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12\!\cdots\!56 \nu^{15} + \cdots - 97\!\cdots\!53 ) / 41\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 23\!\cdots\!21 \nu^{15} + \cdots - 18\!\cdots\!99 ) / 58\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 64\!\cdots\!24 \nu^{15} + \cdots - 31\!\cdots\!12 ) / 13\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 70\!\cdots\!16 \nu^{15} + \cdots + 71\!\cdots\!82 ) / 13\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 40\!\cdots\!49 \nu^{15} + \cdots + 32\!\cdots\!95 ) / 58\!\cdots\!72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} - \beta_{13} + 2\beta_{11} + 2\beta_{9} - 2\beta_{7} + 4\beta_{4} - 2\beta _1 + 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{15} + \beta_{14} - \beta_{13} - 5\beta_{12} + 2\beta_{11} + 8\beta_{9} - 2\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{15} + 3 \beta_{14} - 9 \beta_{13} - 15 \beta_{12} + 27 \beta_{11} + 6 \beta_{10} + 22 \beta_{9} + \cdots - 50 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 21 \beta_{14} - 21 \beta_{13} + 36 \beta_{7} + 26 \beta_{6} - 12 \beta_{5} - 84 \beta_{4} + \cdots - 170 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 95 \beta_{15} - 154 \beta_{14} - 36 \beta_{13} + 315 \beta_{12} - 367 \beta_{11} - 122 \beta_{10} + \cdots - 686 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 353 \beta_{15} - 129 \beta_{14} + 129 \beta_{13} + 1077 \beta_{12} - 855 \beta_{11} + \cdots + 354 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1820 \beta_{15} + 1077 \beta_{14} + 2563 \beta_{13} + 5404 \beta_{12} - 5298 \beta_{11} - 2088 \beta_{10} + \cdots + 10140 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 5647 \beta_{14} + 5647 \beta_{13} - 9312 \beta_{7} - 5582 \beta_{6} + 5608 \beta_{5} + 21736 \beta_{4} + \cdots + 29458 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 30531 \beta_{15} + 41027 \beta_{14} + 20035 \beta_{13} - 87183 \beta_{12} + 79461 \beta_{11} + \cdots + 154186 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 88774 \beta_{15} + 27876 \beta_{14} - 27876 \beta_{13} - 251022 \beta_{12} + 214273 \beta_{11} + \cdots - 76300 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 488763 \beta_{15} - 332558 \beta_{14} - 644968 \beta_{13} - 1374263 \beta_{12} + 1214349 \beta_{11} + \cdots - 2370518 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 692729 \beta_{14} - 692729 \beta_{13} + 1138758 \beta_{7} + 665474 \beta_{6} - 739988 \beta_{5} + \cdots - 3324200 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 7681986 \beta_{15} - 10061205 \beta_{14} - 5302767 \beta_{13} + 21459594 \beta_{12} - 18715048 \beta_{11} + \cdots - 36617512 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 21551273 \beta_{15} - 6568255 \beta_{14} + 6568255 \beta_{13} + 60100653 \beta_{12} + \cdots + 17910162 \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 119824983 \beta_{15} + 83214987 \beta_{14} + 156434979 \beta_{13} + 333789299 \beta_{12} + \cdots + 566848706 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4100\mathbb{Z}\right)^\times\).

\(n\) \(1477\) \(2051\) \(3901\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2049.1
2.78478 + 2.78478i
2.78478 − 2.78478i
−0.352591 − 0.352591i
−0.352591 + 0.352591i
1.86098 − 1.86098i
1.86098 + 1.86098i
0.0520819 − 0.0520819i
0.0520819 + 0.0520819i
0.947918 + 0.947918i
0.947918 − 0.947918i
−0.860977 − 0.860977i
−0.860977 + 0.860977i
1.35259 − 1.35259i
1.35259 + 1.35259i
−1.78478 + 1.78478i
−1.78478 − 1.78478i
0 −2.72531 0 0 0 3.16041 0 4.42734 0
2049.2 0 −2.72531 0 0 0 3.16041 0 4.42734 0
2049.3 0 −2.47572 0 0 0 −4.45099 0 3.12920 0
2049.4 0 −2.47572 0 0 0 −4.45099 0 3.12920 0
2049.5 0 −1.51327 0 0 0 −1.03343 0 −0.710016 0
2049.6 0 −1.51327 0 0 0 −1.03343 0 −0.710016 0
2049.7 0 −0.391765 0 0 0 2.47639 0 −2.84652 0
2049.8 0 −0.391765 0 0 0 2.47639 0 −2.84652 0
2049.9 0 0.391765 0 0 0 −2.47639 0 −2.84652 0
2049.10 0 0.391765 0 0 0 −2.47639 0 −2.84652 0
2049.11 0 1.51327 0 0 0 1.03343 0 −0.710016 0
2049.12 0 1.51327 0 0 0 1.03343 0 −0.710016 0
2049.13 0 2.47572 0 0 0 4.45099 0 3.12920 0
2049.14 0 2.47572 0 0 0 4.45099 0 3.12920 0
2049.15 0 2.72531 0 0 0 −3.16041 0 4.42734 0
2049.16 0 2.72531 0 0 0 −3.16041 0 4.42734 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2049.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
41.b even 2 1 inner
205.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4100.2.g.e 16
5.b even 2 1 inner 4100.2.g.e 16
5.c odd 4 1 820.2.b.b ✓ 8
5.c odd 4 1 4100.2.b.g 8
15.e even 4 1 7380.2.b.h 8
20.e even 4 1 3280.2.b.l 8
41.b even 2 1 inner 4100.2.g.e 16
205.c even 2 1 inner 4100.2.g.e 16
205.g odd 4 1 820.2.b.b ✓ 8
205.g odd 4 1 4100.2.b.g 8
615.p even 4 1 7380.2.b.h 8
820.r even 4 1 3280.2.b.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
820.2.b.b ✓ 8 5.c odd 4 1
820.2.b.b ✓ 8 205.g odd 4 1
3280.2.b.l 8 20.e even 4 1
3280.2.b.l 8 820.r even 4 1
4100.2.b.g 8 5.c odd 4 1
4100.2.b.g 8 205.g odd 4 1
4100.2.g.e 16 1.a even 1 1 trivial
4100.2.g.e 16 5.b even 2 1 inner
4100.2.g.e 16 41.b even 2 1 inner
4100.2.g.e 16 205.c even 2 1 inner
7380.2.b.h 8 15.e even 4 1
7380.2.b.h 8 615.p even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 16T_{3}^{6} + 79T_{3}^{4} - 116T_{3}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(4100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} - 16 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 37 T^{6} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 52 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 53 T^{6} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 96 T^{6} + \cdots + 104976)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 93 T^{6} + \cdots + 112896)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 106 T^{6} + \cdots + 104976)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 73 T^{6} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 7 T^{3} + \cdots - 1296)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 123 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 6 T^{7} + \cdots + 2825761)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 191 T^{6} + \cdots + 1056784)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 171 T^{6} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 116 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 5 T^{3} + \cdots + 144)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} - 4 T^{3} + \cdots + 3684)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} - 237 T^{6} + \cdots + 501264)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 452 T^{6} + \cdots + 1149184)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 479 T^{6} + \cdots + 169744)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 647 T^{6} + \cdots + 589824)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 247 T^{6} + \cdots + 11664)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 404 T^{6} + \cdots + 78287104)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 532 T^{6} + \cdots + 26873856)^{2} \) Copy content Toggle raw display
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