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Properties

Label 1100.3.e.c
Level $1100$
Weight $3$
Character orbit 1100.e
Analytic conductor $29.973$
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] = [1100,3,Mod(549,1100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1100.549");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1100.e (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: \(29.9728290796\)
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: \( x^{16} + 174 x^{14} + 10969 x^{12} + 318076 x^{10} + 4442560 x^{8} + 28982576 x^{6} + 77210944 x^{4} + \cdots + 26790976 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{6} \)
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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{3} + \beta_{3} q^{7} + (\beta_{5} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{3} + \beta_{3} q^{7} + (\beta_{5} - 2) q^{9} - \beta_{14} q^{11} + \beta_{4} q^{13} + (\beta_{4} - \beta_1) q^{17} + ( - 2 \beta_{12} - \beta_{11}) q^{19} + (\beta_{14} - \beta_{13} + \cdots + \beta_{11}) q^{21}+ \cdots + ( - \beta_{15} + 2 \beta_{14} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{9} + 6 q^{11} + 28 q^{31} - 28 q^{49} - 256 q^{59} + 352 q^{69} - 68 q^{71} - 256 q^{81} + 292 q^{89} + 228 q^{91} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 174 x^{14} + 10969 x^{12} + 318076 x^{10} + 4442560 x^{8} + 28982576 x^{6} + 77210944 x^{4} + \cdots + 26790976 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 526387 \nu^{14} + 203480961 \nu^{12} + 23723854247 \nu^{10} + 1148673656900 \nu^{8} + \cdots + 210670458674872 ) / 2225224419000 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 917766 \nu^{14} + 131484803 \nu^{12} + 5534327756 \nu^{10} + 43204757895 \nu^{8} + \cdots + 4738971201376 ) / 890089767600 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11451029 \nu^{14} + 1983497462 \nu^{12} + 123874586349 \nu^{10} + 3517331479300 \nu^{8} + \cdots + 191173811017824 ) / 8900897676000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 32249977 \nu^{14} + 5676229656 \nu^{12} + 363536149037 \nu^{10} + 10730222257250 \nu^{8} + \cdots + 898397195478112 ) / 8900897676000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18944797 \nu^{14} + 3151498906 \nu^{12} + 184168214637 \nu^{10} + 4691618268860 \nu^{8} + \cdots + 98809955440272 ) / 4450448838000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2662577481 \nu^{15} - 104555473034 \nu^{14} + 585490620928 \nu^{13} - 17382891621452 \nu^{12} + \cdots - 19\!\cdots\!04 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3993693 \nu^{15} - 716892818 \nu^{13} - 47193343976 \nu^{11} - 1440595524246 \nu^{9} + \cdots + 9180524157368 \nu ) / 10470692357040 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2511136919 \nu^{15} + 423459819467 \nu^{13} + 25313925940209 \nu^{11} + \cdots - 17\!\cdots\!76 \nu ) / 57\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 6242458011 \nu^{15} + 1078441235378 \nu^{13} + 67080527025131 \nu^{11} + \cdots + 18\!\cdots\!36 \nu ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 7684851319 \nu^{15} - 1432410259862 \nu^{13} - 99302775830299 \nu^{11} + \cdots - 23\!\cdots\!44 \nu ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 973057963 \nu^{15} + 167343588709 \nu^{13} + 10342555557738 \nu^{11} + \cdots + 24\!\cdots\!68 \nu ) / 575888079637200 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1192711078 \nu^{15} + 206772693699 \nu^{13} + 12938189476418 \nu^{11} + \cdots + 26\!\cdots\!28 \nu ) / 575888079637200 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 65851267209 \nu^{15} - 46828946436 \nu^{14} + 11262642085452 \nu^{13} + \cdots - 37\!\cdots\!76 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 65851267209 \nu^{15} - 46828946436 \nu^{14} - 11262642085452 \nu^{13} + \cdots - 37\!\cdots\!76 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 21306883884 \nu^{15} + 3764891388827 \nu^{13} + 242783403755804 \nu^{11} + \cdots + 95\!\cdots\!04 \nu ) / 57\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{12} - \beta_{11} + \beta_{7} ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -5\beta_{14} - 5\beta_{13} - 5\beta_{5} - 2\beta_{4} + 2\beta_{3} + 5\beta_{2} + 2\beta _1 - 110 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 6 \beta_{15} + 8 \beta_{14} - 8 \beta_{13} - 22 \beta_{12} + 70 \beta_{11} + 15 \beta_{10} + \cdots - 61 \beta_{7} ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 375 \beta_{14} + 375 \beta_{13} + 32 \beta_{10} + 32 \beta_{8} + 64 \beta_{6} + 375 \beta_{5} + \cdots + 5360 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 364 \beta_{15} - 852 \beta_{14} + 852 \beta_{13} + 716 \beta_{12} - 4728 \beta_{11} + \cdots + 4801 \beta_{7} ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 25655 \beta_{14} - 25655 \beta_{13} - 3472 \beta_{10} - 3472 \beta_{8} - 6944 \beta_{6} + \cdots - 321580 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 22280 \beta_{15} + 67840 \beta_{14} - 67840 \beta_{13} - 31452 \beta_{12} + 322992 \beta_{11} + \cdots - 374081 \beta_{7} ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1768975 \beta_{14} + 1768975 \beta_{13} + 294304 \beta_{10} + 294304 \beta_{8} + 588608 \beta_{6} + \cdots + 21118620 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1458328 \beta_{15} - 5021504 \beta_{14} + 5021504 \beta_{13} + 1723356 \beta_{12} + \cdots + 28589041 \beta_{7} ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 124122655 \beta_{14} - 124122655 \beta_{13} - 23040344 \beta_{10} - 23040344 \beta_{8} + \cdots - 1453108380 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 100170904 \beta_{15} + 364417472 \beta_{14} - 364417472 \beta_{13} - 108506532 \beta_{12} + \cdots - 2149794241 \beta_{7} ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 8832108175 \beta_{14} + 8832108175 \beta_{13} + 1743155536 \beta_{10} + 1743155536 \beta_{8} + \cdots + 102566644860 ) / 5 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 7074595592 \beta_{15} - 26354449056 \beta_{14} + 26354449056 \beta_{13} + 7367236796 \beta_{12} + \cdots + 159833110321 \beta_{7} ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 634462328255 \beta_{14} - 634462328255 \beta_{13} - 129693803336 \beta_{10} - 129693803336 \beta_{8} + \cdots - 7343348641660 ) / 5 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 507170631128 \beta_{15} + 1908289559904 \beta_{14} - 1908289559904 \beta_{13} + \cdots - 11795987667041 \beta_{7} ) / 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(551\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
549.1
3.45778i
− 5.45778i
1.23882i
0.761183i
− 8.54086i
6.54086i
3.49129i
− 1.49129i
− 3.49129i
1.49129i
8.54086i
− 6.54086i
− 1.23882i
− 0.761183i
− 3.45778i
5.45778i
0 − 4.69333i 0 0 0 −7.54848 0 −13.0273 0
549.2 0 − 4.69333i 0 0 0 7.54848 0 −13.0273 0
549.3 0 − 4.14607i 0 0 0 −1.70661 0 −8.18991 0
549.4 0 − 4.14607i 0 0 0 1.70661 0 −8.18991 0
549.5 0 − 2.09157i 0 0 0 −6.85033 0 4.62533 0
549.6 0 − 2.09157i 0 0 0 6.85033 0 4.62533 0
549.7 0 − 0.638826i 0 0 0 −9.06537 0 8.59190 0
549.8 0 − 0.638826i 0 0 0 9.06537 0 8.59190 0
549.9 0 0.638826i 0 0 0 −9.06537 0 8.59190 0
549.10 0 0.638826i 0 0 0 9.06537 0 8.59190 0
549.11 0 2.09157i 0 0 0 −6.85033 0 4.62533 0
549.12 0 2.09157i 0 0 0 6.85033 0 4.62533 0
549.13 0 4.14607i 0 0 0 −1.70661 0 −8.18991 0
549.14 0 4.14607i 0 0 0 1.70661 0 −8.18991 0
549.15 0 4.69333i 0 0 0 −7.54848 0 −13.0273 0
549.16 0 4.69333i 0 0 0 7.54848 0 −13.0273 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 549.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.b odd 2 1 inner
55.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1100.3.e.c 16
5.b even 2 1 inner 1100.3.e.c 16
5.c odd 4 1 1100.3.f.c ✓ 8
5.c odd 4 1 1100.3.f.e yes 8
11.b odd 2 1 inner 1100.3.e.c 16
55.d odd 2 1 inner 1100.3.e.c 16
55.e even 4 1 1100.3.f.c ✓ 8
55.e even 4 1 1100.3.f.e yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1100.3.e.c 16 1.a even 1 1 trivial
1100.3.e.c 16 5.b even 2 1 inner
1100.3.e.c 16 11.b odd 2 1 inner
1100.3.e.c 16 55.d odd 2 1 inner
1100.3.f.c ✓ 8 5.c odd 4 1
1100.3.f.c ✓ 8 55.e even 4 1
1100.3.f.e yes 8 5.c odd 4 1
1100.3.f.e yes 8 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 44T_{3}^{6} + 568T_{3}^{4} + 1881T_{3}^{2} + 676 \) acting on \(S_{3}^{\mathrm{new}}(1100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 44 T^{6} + \cdots + 676)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 189 T^{6} + \cdots + 640000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 3 T^{7} + \cdots + 214358881)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 461 T^{6} + \cdots + 1562500)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 1854 T^{6} + \cdots + 2560000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 2660 T^{6} + \cdots + 71355765625)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 1432 T^{6} + \cdots + 2633947684)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 3996 T^{6} + \cdots + 125139062500)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 7 T^{3} - 1135 T^{2} + \cdots - 46)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 7318 T^{6} + \cdots + 295840000)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 7715 T^{6} + \cdots + 175561000000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 9721 T^{6} + \cdots + 207571360000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 5186 T^{6} + \cdots + 2970250000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 9515 T^{6} + \cdots + 13386490000)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 64 T^{3} + \cdots - 8018420)^{4} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 20675209000000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 10144377880576)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 17 T^{3} + \cdots - 250696)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 50809096802500)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 121475666560000)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 57826519140625)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 73 T^{3} + \cdots - 5154671)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 6428303868100)^{2} \) Copy content Toggle raw display
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