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Properties

Label 1100.3.e.b
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: \( x^{16} + 85x^{14} + 2456x^{12} + 32605x^{10} + 215801x^{8} + 712960x^{6} + 1098976x^{4} + 633600x^{2} + 92416 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{25}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 220)
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.

\(f(q)\) \(=\) \( q + (\beta_{10} - \beta_{9}) q^{3} - \beta_{4} q^{7} + (\beta_{2} + \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{10} - \beta_{9}) q^{3} - \beta_{4} q^{7} + (\beta_{2} + \beta_1 - 3) q^{9} + ( - \beta_{13} + \beta_{3} + 3) q^{11} + \beta_{7} q^{13} + (\beta_{6} + \beta_{4}) q^{17} + \beta_{12} q^{19} + (\beta_{15} - \beta_{14} + \cdots - \beta_1) q^{21}+ \cdots + ( - 6 \beta_{15} + \beta_{14} + \cdots + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 56 q^{9} + 48 q^{11} + 40 q^{31} + 488 q^{49} + 32 q^{59} + 112 q^{69} - 440 q^{71} - 448 q^{81} - 440 q^{89} - 144 q^{91} + 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 85x^{14} + 2456x^{12} + 32605x^{10} + 215801x^{8} + 712960x^{6} + 1098976x^{4} + 633600x^{2} + 92416 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 773771 \nu^{14} + 63432595 \nu^{12} + 1709306672 \nu^{10} + 20113525535 \nu^{8} + \cdots + 87809107120 ) / 5355322600 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 420887 \nu^{14} + 34184767 \nu^{12} + 903498960 \nu^{10} + 10228121691 \nu^{8} + \cdots - 11318737840 ) / 1071064520 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2955 \nu^{14} + 242607 \nu^{12} + 6556704 \nu^{10} + 77544527 \nu^{8} + 417992347 \nu^{6} + \cdots + 146115456 ) / 5824960 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 54764265 \nu^{14} + 4482938619 \nu^{12} + 120417116030 \nu^{10} + 1407174092933 \nu^{8} + \cdots + 788722334496 ) / 85685161600 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 30071229 \nu^{14} + 2488784233 \nu^{12} + 68283056336 \nu^{10} + 827610903545 \nu^{8} + \cdots + 1644909664896 ) / 34274064640 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 108529015 \nu^{14} + 8861050787 \nu^{12} + 236701101560 \nu^{10} + 2733874494699 \nu^{8} + \cdots + 217449787648 ) / 85685161600 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 73311950 \nu^{14} - 6017693411 \nu^{12} - 162514662625 \nu^{10} - 1917452946702 \nu^{8} + \cdots - 3688089010224 ) / 42842580800 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1063643 \nu^{15} + 89743059 \nu^{13} + 2558264900 \nu^{11} + 33247815327 \nu^{9} + \cdots + 318422512960 \nu ) / 15597777920 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13546063 \nu^{15} + 1108359607 \nu^{13} + 29739874116 \nu^{11} + 346608435459 \nu^{9} + \cdots - 234099532992 \nu ) / 119487564800 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 481943745 \nu^{15} - 39753406153 \nu^{13} - 1084368718700 \nu^{11} + \cdots - 70628062332352 \nu ) / 3256036140800 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2012005689 \nu^{15} + 173225094529 \nu^{13} + 5114270628428 \nu^{11} + \cdots + 706369897354176 \nu ) / 13024144563200 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 212050611 \nu^{15} - 17700040563 \nu^{13} - 494121743628 \nu^{11} + \cdots - 44068563980224 \nu ) / 651207228160 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 777460604 \nu^{15} + 295333283 \nu^{14} + 63102564522 \nu^{13} + 24261360775 \nu^{12} + \cdots + 7071920116480 ) / 1628018070400 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1788796429 \nu^{15} + 590666566 \nu^{14} + 148403486737 \nu^{13} + 48522721550 \nu^{12} + \cdots + 14143840232960 ) / 3256036140800 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 861011933 \nu^{15} - 70769220879 \nu^{13} - 1916537290046 \nu^{11} + \cdots - 31300004371616 \nu ) / 814009035200 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - \beta_{14} - 3 \beta_{13} - 7 \beta_{12} - 5 \beta_{11} - 5 \beta_{10} + 10 \beta_{9} + \cdots - 2 \beta_1 ) / 40 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6\beta_{7} + \beta_{6} + 6\beta_{5} - 50\beta_{4} + 40\beta_{3} + 30\beta_{2} + 25\beta _1 - 410 ) / 40 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 41 \beta_{15} + 21 \beta_{14} + 63 \beta_{13} + 67 \beta_{12} + 75 \beta_{11} + 155 \beta_{10} + \cdots + 42 \beta_1 ) / 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -110\beta_{7} - 7\beta_{6} - 82\beta_{5} + 386\beta_{4} - 392\beta_{3} - 294\beta_{2} - 275\beta _1 + 2166 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4271 \beta_{15} - 2051 \beta_{14} - 4473 \beta_{13} - 4437 \beta_{12} - 5335 \beta_{11} + \cdots - 3262 \beta_1 ) / 40 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3442 \beta_{7} + 222 \beta_{6} + 2432 \beta_{5} - 9770 \beta_{4} + 10640 \beta_{3} + 7955 \beta_{2} + \cdots - 49560 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 192517 \beta_{15} + 93517 \beta_{14} + 176511 \beta_{13} + 174539 \beta_{12} + 213745 \beta_{11} + \cdots + 135014 \beta_1 ) / 40 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 248086 \beta_{7} - 17505 \beta_{6} - 172070 \beta_{5} + 656962 \beta_{4} - 732664 \beta_{3} + \cdots + 3236314 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4190913 \beta_{15} - 2056693 \beta_{14} - 3680799 \beta_{13} - 3635731 \beta_{12} + \cdots - 2868746 \beta_1 ) / 20 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 54199166 \beta_{7} + 4013351 \beta_{6} + 37312306 \beta_{5} - 140116370 \beta_{4} + 157640200 \beta_{3} + \cdots - 684094550 ) / 40 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 361683107 \beta_{15} + 178409047 \beta_{14} + 313416661 \beta_{13} + 309151329 \beta_{12} + \cdots + 245912854 \beta_1 ) / 40 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 58606476 \beta_{7} - 4432468 \beta_{6} - 40222208 \beta_{5} + 150227036 \beta_{4} - 169581712 \beta_{3} + \cdots + 731388184 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 15570307241 \beta_{15} - 7697637281 \beta_{14} - 13436291003 \beta_{13} - 13241136407 \beta_{12} + \cdots - 10566964142 \beta_1 ) / 40 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 101036123446 \beta_{7} + 7707866201 \beta_{6} + 69255248406 \beta_{5} - 258201715330 \beta_{4} + \cdots - 1255915958570 ) / 40 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 334940993401 \beta_{15} + 165739614741 \beta_{14} + 288659810863 \beta_{13} + \cdots + 227199712802 \beta_1 ) / 20 \) 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
6.55851i
− 1.78705i
− 3.95808i
0.465196i
− 2.61274i
1.73790i
3.50416i
0.885333i
− 3.50416i
− 0.885333i
2.61274i
− 1.73790i
3.95808i
− 0.465196i
− 6.55851i
1.78705i
0 − 4.94892i 0 0 0 −13.1585 0 −15.4918 0
549.2 0 − 4.94892i 0 0 0 13.1585 0 −15.4918 0
549.3 0 − 3.65160i 0 0 0 −9.09184 0 −4.33416 0
549.4 0 − 3.65160i 0 0 0 9.09184 0 −4.33416 0
549.5 0 − 3.41553i 0 0 0 −0.558647 0 −2.66584 0
549.6 0 − 3.41553i 0 0 0 0.558647 0 −2.66584 0
549.7 0 − 0.712855i 0 0 0 −7.86633 0 8.49184 0
549.8 0 − 0.712855i 0 0 0 7.86633 0 8.49184 0
549.9 0 0.712855i 0 0 0 −7.86633 0 8.49184 0
549.10 0 0.712855i 0 0 0 7.86633 0 8.49184 0
549.11 0 3.41553i 0 0 0 −0.558647 0 −2.66584 0
549.12 0 3.41553i 0 0 0 0.558647 0 −2.66584 0
549.13 0 3.65160i 0 0 0 −9.09184 0 −4.33416 0
549.14 0 3.65160i 0 0 0 9.09184 0 −4.33416 0
549.15 0 4.94892i 0 0 0 −13.1585 0 −15.4918 0
549.16 0 4.94892i 0 0 0 13.1585 0 −15.4918 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.b 16
5.b even 2 1 inner 1100.3.e.b 16
5.c odd 4 1 220.3.f.a ✓ 8
5.c odd 4 1 1100.3.f.f 8
11.b odd 2 1 inner 1100.3.e.b 16
15.e even 4 1 1980.3.b.a 8
20.e even 4 1 880.3.j.b 8
55.d odd 2 1 inner 1100.3.e.b 16
55.e even 4 1 220.3.f.a ✓ 8
55.e even 4 1 1100.3.f.f 8
165.l odd 4 1 1980.3.b.a 8
220.i odd 4 1 880.3.j.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.3.f.a ✓ 8 5.c odd 4 1
220.3.f.a ✓ 8 55.e even 4 1
880.3.j.b 8 20.e even 4 1
880.3.j.b 8 220.i odd 4 1
1100.3.e.b 16 1.a even 1 1 trivial
1100.3.e.b 16 5.b even 2 1 inner
1100.3.e.b 16 11.b odd 2 1 inner
1100.3.e.b 16 55.d odd 2 1 inner
1100.3.f.f 8 5.c odd 4 1
1100.3.f.f 8 55.e even 4 1
1980.3.b.a 8 15.e even 4 1
1980.3.b.a 8 165.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 50T_{3}^{6} + 793T_{3}^{4} + 4200T_{3}^{2} + 1936 \) 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
$3$ \( (T^{8} + 50 T^{6} + \cdots + 1936)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 318 T^{6} + \cdots + 276400)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 24 T^{7} + \cdots + 214358881)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 1052 T^{6} + \cdots + 110560000)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 1878 T^{6} + \cdots + 12073428400)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 650 T^{6} + \cdots + 110560000)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 1336 T^{6} + \cdots + 495616)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 4002 T^{6} + \cdots + 100835142400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 10 T^{3} + \cdots + 1062716)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 2938 T^{6} + \cdots + 1341024400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 7245660160000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 5572 T^{6} + \cdots + 840680550400)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 5912 T^{6} + \cdots + 2560000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 6410 T^{6} + \cdots + 32921925136)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 8 T^{3} + \cdots - 735680)^{4} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 75981364960000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 116743357286656)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 110 T^{3} + \cdots - 24870124)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} - 27932 T^{6} + \cdots + 22293318400)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 110 T^{3} + \cdots + 44004556)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 60908659360000)^{2} \) Copy content Toggle raw display
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