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Properties

Label 1104.3.g.c
Level $1104$
Weight $3$
Character orbit 1104.g
Analytic conductor $30.082$
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] = [1104,3,Mod(737,1104)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1104, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1104.737");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1104.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: \(30.0818211854\)
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: \( x^{16} - 4 x^{15} + 10 x^{14} + 8 x^{13} - 119 x^{12} + 416 x^{11} - 774 x^{10} - 1284 x^{9} + \cdots + 43046721 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 138)
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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{3} - \beta_{8} q^{5} + \beta_{4} q^{7} + (\beta_{13} + \beta_{11} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{3} - \beta_{8} q^{5} + \beta_{4} q^{7} + (\beta_{13} + \beta_{11} - \beta_{2}) q^{9} + ( - \beta_{13} + \beta_{12} + \cdots + \beta_1) q^{11}+ \cdots + ( - 2 \beta_{15} + \beta_{14} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{3} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{3} - 4 q^{9} - 8 q^{13} - 28 q^{15} - 40 q^{19} + 4 q^{21} - 192 q^{25} + 80 q^{27} - 136 q^{31} - 84 q^{33} - 136 q^{37} - 156 q^{39} - 72 q^{43} + 4 q^{45} + 224 q^{49} + 4 q^{51} + 96 q^{55} - 160 q^{57} + 48 q^{61} - 204 q^{63} + 304 q^{67} + 408 q^{73} - 68 q^{75} - 312 q^{79} + 164 q^{81} - 464 q^{85} + 268 q^{87} + 72 q^{91} - 108 q^{93} + 168 q^{97} + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 10 x^{14} + 8 x^{13} - 119 x^{12} + 416 x^{11} - 774 x^{10} - 1284 x^{9} + \cdots + 43046721 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 851 \nu^{15} - 7976 \nu^{14} + 41297 \nu^{13} - 91076 \nu^{12} - 534502 \nu^{11} + \cdots - 31175391942 ) / 17218688400 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1196 \nu^{15} + 229 \nu^{14} + 31922 \nu^{13} + 135109 \nu^{12} - 473362 \nu^{11} + \cdots - 48035357667 ) / 17218688400 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 157 \nu^{15} - 1249 \nu^{14} - 2993 \nu^{13} + 9383 \nu^{12} - 84644 \nu^{11} + \cdots - 4883411349 ) / 1147912560 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 329 \nu^{15} - 1708 \nu^{14} + 3703 \nu^{13} + 2120 \nu^{12} - 22220 \nu^{11} + \cdots - 10005971148 ) / 2295825120 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 943 \nu^{15} + 3122 \nu^{14} + 1861 \nu^{13} - 44368 \nu^{12} + 194764 \nu^{11} + \cdots + 43171078194 ) / 5739562800 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2876 \nu^{15} - 5119 \nu^{14} + 12298 \nu^{13} + 92561 \nu^{12} + 48232 \nu^{11} + \cdots - 98323493733 ) / 17218688400 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1049 \nu^{15} + 2761 \nu^{14} + 563 \nu^{13} - 46859 \nu^{12} + 119312 \nu^{11} + \cdots + 31342795857 ) / 5739562800 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 6992 \nu^{15} + 36013 \nu^{14} - 58996 \nu^{13} + 326953 \nu^{12} - 578104 \nu^{11} + \cdots + 137983872681 ) / 34437376800 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{15} - 4 \nu^{14} + 10 \nu^{13} + 8 \nu^{12} - 119 \nu^{11} + 416 \nu^{10} - 774 \nu^{9} + \cdots - 19131876 ) / 4782969 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 223 \nu^{15} - 2713 \nu^{14} + 5815 \nu^{13} - 3817 \nu^{12} - 32168 \nu^{11} + \cdots - 8046548181 ) / 765275040 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 6247 \nu^{15} + 15502 \nu^{14} - 37729 \nu^{13} - 59948 \nu^{12} + 164414 \nu^{11} + \cdots + 10120762404 ) / 17218688400 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 6553 \nu^{15} + 35653 \nu^{14} - 40681 \nu^{13} - 47357 \nu^{12} + 358076 \nu^{11} + \cdots + 93645750051 ) / 17218688400 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 773 \nu^{15} + 1103 \nu^{14} + 6139 \nu^{13} - 27127 \nu^{12} + 90736 \nu^{11} + \cdots + 5017334481 ) / 1913187600 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 7336 \nu^{15} - 37651 \nu^{14} + 2422 \nu^{13} + 139319 \nu^{12} - 1257932 \nu^{11} + \cdots - 178275603537 ) / 17218688400 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 9107 \nu^{15} - 18887 \nu^{14} - 21781 \nu^{13} + 179173 \nu^{12} - 627424 \nu^{11} + \cdots - 102121171119 ) / 11479125600 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{14} - \beta_{13} + \beta_{9} + \beta_{6} - \beta_{2} + \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{12} - 3\beta_{11} + 3\beta_{9} - 2\beta_{7} - 3\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{15} + 6 \beta_{14} + 6 \beta_{13} - 6 \beta_{12} - 3 \beta_{11} - 3 \beta_{10} - 6 \beta_{9} + \cdots - 15 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 18 \beta_{14} + 9 \beta_{13} - 6 \beta_{12} + 15 \beta_{11} - 6 \beta_{9} - 2 \beta_{7} - 36 \beta_{6} + \cdots + 36 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 48 \beta_{15} + 72 \beta_{14} - 18 \beta_{13} - 60 \beta_{12} + 60 \beta_{11} - 12 \beta_{10} + \cdots - 18 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 126 \beta_{15} - 144 \beta_{14} + 144 \beta_{13} + 66 \beta_{12} - 66 \beta_{11} + 90 \beta_{10} + \cdots + 171 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 132 \beta_{15} - 81 \beta_{14} + 279 \beta_{13} - 150 \beta_{12} - 732 \beta_{11} - 552 \beta_{10} + \cdots + 1143 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 792 \beta_{15} - 2520 \beta_{14} + 684 \beta_{13} - 1779 \beta_{12} - 147 \beta_{11} - 1656 \beta_{10} + \cdots + 4410 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1119 \beta_{15} - 594 \beta_{14} - 2916 \beta_{13} + 84 \beta_{12} - 3765 \beta_{11} - 1119 \beta_{10} + \cdots - 19611 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 8424 \beta_{15} + 4932 \beta_{14} + 13563 \beta_{13} + 10092 \beta_{12} + 5415 \beta_{11} + 4536 \beta_{10} + \cdots + 48744 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 19176 \beta_{15} - 16416 \beta_{14} - 11268 \beta_{13} - 28896 \beta_{12} + 16368 \beta_{11} + \cdots + 16092 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 36756 \beta_{15} - 8568 \beta_{14} - 78264 \beta_{13} - 48804 \beta_{12} - 7356 \beta_{11} + \cdots + 723663 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 233904 \beta_{15} - 303237 \beta_{14} + 11619 \beta_{13} - 146220 \beta_{12} - 325488 \beta_{11} + \cdots - 852111 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 881640 \beta_{15} - 457416 \beta_{14} + 903024 \beta_{13} + 2298027 \beta_{12} - 1070319 \beta_{11} + \cdots - 3112308 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 142305 \beta_{15} + 1605150 \beta_{14} + 1345626 \beta_{13} - 4899066 \beta_{12} - 1026579 \beta_{11} + \cdots - 16390251 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1104\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(277\) \(415\) \(737\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
737.1
2.97243 − 0.405752i
2.97243 + 0.405752i
2.42141 − 1.77110i
2.42141 + 1.77110i
1.62763 − 2.52009i
1.62763 + 2.52009i
1.30755 − 2.70006i
1.30755 + 2.70006i
−0.0515370 − 2.99956i
−0.0515370 + 2.99956i
−0.752365 − 2.90413i
−0.752365 + 2.90413i
−2.52762 − 1.61590i
−2.52762 + 1.61590i
−2.99749 − 0.122585i
−2.99749 + 0.122585i
0 −2.97243 − 0.405752i 0 6.32168i 0 −5.04690 0 8.67073 + 2.41214i 0
737.2 0 −2.97243 + 0.405752i 0 − 6.32168i 0 −5.04690 0 8.67073 − 2.41214i 0
737.3 0 −2.42141 − 1.77110i 0 4.98213i 0 12.5950 0 2.72641 + 8.57710i 0
737.4 0 −2.42141 + 1.77110i 0 − 4.98213i 0 12.5950 0 2.72641 − 8.57710i 0
737.5 0 −1.62763 − 2.52009i 0 1.81026i 0 −9.47484 0 −3.70167 + 8.20351i 0
737.6 0 −1.62763 + 2.52009i 0 − 1.81026i 0 −9.47484 0 −3.70167 − 8.20351i 0
737.7 0 −1.30755 − 2.70006i 0 − 8.14461i 0 −7.21035 0 −5.58060 + 7.06094i 0
737.8 0 −1.30755 + 2.70006i 0 8.14461i 0 −7.21035 0 −5.58060 − 7.06094i 0
737.9 0 0.0515370 − 2.99956i 0 2.42657i 0 2.87957 0 −8.99469 − 0.309176i 0
737.10 0 0.0515370 + 2.99956i 0 − 2.42657i 0 2.87957 0 −8.99469 + 0.309176i 0
737.11 0 0.752365 − 2.90413i 0 − 9.12709i 0 11.5474 0 −7.86789 − 4.36992i 0
737.12 0 0.752365 + 2.90413i 0 9.12709i 0 11.5474 0 −7.86789 + 4.36992i 0
737.13 0 2.52762 − 1.61590i 0 7.30416i 0 0.709880 0 3.77777 − 8.16875i 0
737.14 0 2.52762 + 1.61590i 0 − 7.30416i 0 0.709880 0 3.77777 + 8.16875i 0
737.15 0 2.99749 − 0.122585i 0 − 4.36583i 0 −5.99969 0 8.96995 − 0.734895i 0
737.16 0 2.99749 + 0.122585i 0 4.36583i 0 −5.99969 0 8.96995 + 0.734895i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 737.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1104.3.g.c 16
3.b odd 2 1 inner 1104.3.g.c 16
4.b odd 2 1 138.3.c.a ✓ 16
12.b even 2 1 138.3.c.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.3.c.a ✓ 16 4.b odd 2 1
138.3.c.a ✓ 16 12.b even 2 1
1104.3.g.c 16 1.a even 1 1 trivial
1104.3.g.c 16 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 296 T_{5}^{14} + 35404 T_{5}^{12} + 2204176 T_{5}^{10} + 76670644 T_{5}^{8} + \cdots + 107557761600 \) acting on \(S_{3}^{\mathrm{new}}(1104, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 4 T^{15} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 107557761600 \) Copy content Toggle raw display
$7$ \( (T^{8} - 252 T^{6} + \cdots + 615000)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 429981696 \) Copy content Toggle raw display
$13$ \( (T^{8} + 4 T^{7} + \cdots + 960369700)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( (T^{8} + 20 T^{7} + \cdots - 780768)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 23)^{8} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} + 68 T^{7} + \cdots + 145917216)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 68 T^{7} + \cdots + 44098389696)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 69\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 2868882576000)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 30\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 20431830161152)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 3918307649688)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 26\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 171604877785344)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 173075158110240)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{8} - 84 T^{7} + \cdots - 44369259840)^{2} \) Copy content Toggle raw display
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