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Properties

Label 4732.2.g.l
Level $4732$
Weight $2$
Character orbit 4732.g
Analytic conductor $37.785$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4732,2,Mod(337,4732)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4732, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4732.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4732 = 2^{2} \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4732.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: \(37.7852102365\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 30 x^{16} + 361 x^{14} + 2268 x^{12} + 8158 x^{10} + 17375 x^{8} + 21892 x^{6} + 15714 x^{4} + \cdots + 841 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_1 - 1) q^{3} - \beta_{12} q^{5} + \beta_{14} q^{7} + ( - \beta_{7} + \beta_{6} - 2 \beta_{4} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + \beta_1 - 1) q^{3} - \beta_{12} q^{5} + \beta_{14} q^{7} + ( - \beta_{7} + \beta_{6} - 2 \beta_{4} + \cdots + 2) q^{9}+ \cdots + ( - 2 \beta_{17} - 6 \beta_{16} + \cdots - 3 \beta_{9}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 12 q^{3} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 12 q^{3} + 14 q^{9} - 2 q^{17} + 54 q^{23} - 14 q^{25} - 60 q^{27} - 28 q^{29} - 8 q^{35} + 38 q^{43} - 18 q^{49} + 20 q^{51} - 60 q^{53} - 4 q^{55} - 24 q^{61} - 38 q^{69} + 54 q^{75} + 10 q^{77} + 2 q^{79} + 106 q^{81} - 62 q^{87} + 82 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 30 x^{16} + 361 x^{14} + 2268 x^{12} + 8158 x^{10} + 17375 x^{8} + 21892 x^{6} + 15714 x^{4} + \cdots + 841 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 18074 \nu^{16} + 499216 \nu^{14} + 5331129 \nu^{12} + 28159401 \nu^{10} + 79083418 \nu^{8} + \cdots + 13231530 ) / 1388103 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 53294 \nu^{16} + 1540881 \nu^{14} + 17599037 \nu^{12} + 102675892 \nu^{10} + 332740464 \nu^{8} + \cdots + 62678256 ) / 1388103 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 227390 \nu^{16} + 6569334 \nu^{14} + 74791328 \nu^{12} + 432560140 \nu^{10} + 1373425047 \nu^{8} + \cdots + 178959696 ) / 1388103 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1205 \nu^{16} - 34612 \nu^{14} - 390836 \nu^{12} - 2234252 \nu^{10} - 6979460 \nu^{8} + \cdots - 792512 ) / 4119 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 451051 \nu^{16} + 12887086 \nu^{14} + 144358450 \nu^{12} + 815079675 \nu^{10} + 2496973964 \nu^{8} + \cdots + 207391970 ) / 1388103 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 552994 \nu^{16} + 15869632 \nu^{14} + 178960661 \nu^{12} + 1021075363 \nu^{10} + 3181268473 \nu^{8} + \cdots + 354770793 ) / 1388103 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 780384 \nu^{16} - 22438966 \nu^{14} - 253751989 \nu^{12} - 1453635503 \nu^{10} + \cdots - 528178077 ) / 1388103 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1179613 \nu^{16} + 33908057 \nu^{14} + 383293548 \nu^{12} + 2194532955 \nu^{10} + 6871581989 \nu^{8} + \cdots + 810681756 ) / 1388103 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 53294 \nu^{17} - 1540881 \nu^{15} - 17599037 \nu^{13} - 102675892 \nu^{11} + \cdots - 64066359 \nu ) / 1388103 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2937652 \nu^{17} - 83470217 \nu^{15} - 926267235 \nu^{13} - 5141341011 \nu^{11} + \cdots + 125054406 \nu ) / 40254987 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3038512 \nu^{17} - 85973205 \nu^{15} - 949150442 \nu^{13} - 5240653500 \nu^{11} + \cdots - 600161017 \nu ) / 40254987 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 7313014 \nu^{17} - 208635335 \nu^{15} - 2331943263 \nu^{13} - 13122044856 \nu^{11} + \cdots - 2976304338 \nu ) / 40254987 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 8685390 \nu^{17} + 250032351 \nu^{15} + 2831776679 \nu^{13} + 16250964791 \nu^{11} + \cdots + 5747778884 \nu ) / 40254987 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 27328 \nu^{17} + 784895 \nu^{15} + 8861660 \nu^{13} + 50645660 \nu^{11} + 158148516 \nu^{9} + \cdots + 18194342 \nu ) / 119451 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 12185610 \nu^{17} - 349531474 \nu^{15} - 3938785882 \nu^{13} - 22447104311 \nu^{11} + \cdots - 7369433688 \nu ) / 40254987 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 12185610 \nu^{17} + 349531474 \nu^{15} + 3938785882 \nu^{13} + 22447104311 \nu^{11} + \cdots + 7409688675 \nu ) / 40254987 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 14830763 \nu^{17} - 425057484 \nu^{15} - 4783728700 \nu^{13} - 27206439225 \nu^{11} + \cdots - 9175049945 \nu ) / 40254987 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{16} + \beta_{15} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -5\beta_{16} - 6\beta_{15} + 2\beta_{14} - 2\beta_{13} + 2\beta_{12} - \beta_{10} + \beta_{9} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{8} - 8\beta_{7} - 11\beta_{6} - 11\beta_{3} - \beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{17} + 33 \beta_{16} + 45 \beta_{15} - 26 \beta_{14} + 25 \beta_{13} - 19 \beta_{12} + \cdots - 13 \beta_{9} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -27\beta_{8} + 63\beta_{7} + 102\beta_{6} - 2\beta_{5} - 5\beta_{4} + 103\beta_{3} - 5\beta_{2} + 18\beta _1 - 217 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 34 \beta_{17} - 248 \beta_{16} - 360 \beta_{15} + 287 \beta_{14} - 255 \beta_{13} + 165 \beta_{12} + \cdots + 141 \beta_{9} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 289 \beta_{8} - 514 \beta_{7} - 921 \beta_{6} + 40 \beta_{5} + 103 \beta_{4} - 923 \beta_{3} + \cdots + 1849 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 432 \beta_{17} + 1985 \beta_{16} + 2984 \beta_{15} - 2955 \beta_{14} + 2446 \beta_{13} + \cdots - 1442 \beta_{9} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2878 \beta_{8} + 4303 \beta_{7} + 8307 \beta_{6} - 559 \beta_{5} - 1423 \beta_{4} + 8161 \beta_{3} + \cdots - 16162 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4860 \beta_{17} - 16443 \beta_{16} - 25317 \beta_{15} + 29278 \beta_{14} - 22911 \beta_{13} + \cdots + 14311 \beta_{9} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 27771 \beta_{8} - 36655 \beta_{7} - 75149 \beta_{6} + 6709 \beta_{5} + 16690 \beta_{4} - 72006 \beta_{3} + \cdots + 143232 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 51170 \beta_{17} + 139115 \beta_{16} + 218278 \beta_{15} - 283434 \beta_{14} + \cdots - 139320 \beta_{9} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 263536 \beta_{8} + 316216 \beta_{7} + 681768 \beta_{6} - 74023 \beta_{5} - 179869 \beta_{4} + \cdots - 1280076 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 517428 \beta_{17} - 1194237 \beta_{16} - 1903998 \beta_{15} + 2703155 \beta_{14} + \cdots + 1337715 \beta_{9} \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 2475872 \beta_{8} - 2755008 \beta_{7} - 6198141 \beta_{6} + 774928 \beta_{5} + 1844947 \beta_{4} + \cdots + 11503657 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 5095747 \beta_{17} + 10365275 \beta_{16} + 16755749 \beta_{15} - 25519248 \beta_{14} + \cdots - 12711406 \beta_{9} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4732\mathbb{Z}\right)^\times\).

\(n\) \(2367\) \(2705\) \(4565\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.11607i
− 1.11607i
− 2.72084i
2.72084i
0.794150i
− 0.794150i
− 2.13429i
2.13429i
1.14100i
− 1.14100i
0.634469i
− 0.634469i
− 3.02399i
3.02399i
1.82297i
− 1.82297i
1.41186i
− 1.41186i
0 −3.36305 0 − 2.18646i 0 1.00000i 0 8.31012 0
337.2 0 −3.36305 0 2.18646i 0 − 1.00000i 0 8.31012 0
337.3 0 −3.27580 0 − 3.07655i 0 − 1.00000i 0 7.73088 0
337.4 0 −3.27580 0 3.07655i 0 1.00000i 0 7.73088 0
337.5 0 −1.34911 0 − 0.413269i 0 1.00000i 0 −1.17991 0
337.6 0 −1.34911 0 0.413269i 0 − 1.00000i 0 −1.17991 0
337.7 0 −1.33235 0 − 1.52015i 0 − 1.00000i 0 −1.22484 0
337.8 0 −1.33235 0 1.52015i 0 1.00000i 0 −1.22484 0
337.9 0 −1.10598 0 − 4.06755i 0 − 1.00000i 0 −1.77680 0
337.10 0 −1.10598 0 4.06755i 0 1.00000i 0 −1.77680 0
337.11 0 0.167469 0 − 3.79534i 0 1.00000i 0 −2.97195 0
337.12 0 0.167469 0 3.79534i 0 − 1.00000i 0 −2.97195 0
337.13 0 0.777013 0 − 0.436047i 0 1.00000i 0 −2.39625 0
337.14 0 0.777013 0 0.436047i 0 − 1.00000i 0 −2.39625 0
337.15 0 1.26801 0 − 0.138655i 0 − 1.00000i 0 −1.39214 0
337.16 0 1.26801 0 0.138655i 0 1.00000i 0 −1.39214 0
337.17 0 2.21380 0 − 2.02821i 0 − 1.00000i 0 1.90090 0
337.18 0 2.21380 0 2.02821i 0 1.00000i 0 1.90090 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.18
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4732.2.g.l 18
13.b even 2 1 inner 4732.2.g.l 18
13.d odd 4 1 4732.2.a.u ✓ 9
13.d odd 4 1 4732.2.a.v yes 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4732.2.a.u ✓ 9 13.d odd 4 1
4732.2.a.v yes 9 13.d odd 4 1
4732.2.g.l 18 1.a even 1 1 trivial
4732.2.g.l 18 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4732, [\chi])\):

\( T_{3}^{9} + 6T_{3}^{8} + T_{3}^{7} - 44T_{3}^{6} - 45T_{3}^{5} + 77T_{3}^{4} + 93T_{3}^{3} - 40T_{3}^{2} - 44T_{3} + 8 \) Copy content Toggle raw display
\( T_{5}^{18} + 52 T_{5}^{16} + 1044 T_{5}^{14} + 10271 T_{5}^{12} + 52276 T_{5}^{10} + 133688 T_{5}^{8} + \cdots + 64 \) Copy content Toggle raw display
\( T_{17}^{9} + T_{17}^{8} - 76 T_{17}^{7} - 33 T_{17}^{6} + 1579 T_{17}^{5} - 42 T_{17}^{4} - 11821 T_{17}^{3} + \cdots - 21064 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( (T^{9} + 6 T^{8} + T^{7} + \cdots + 8)^{2} \) Copy content Toggle raw display
$5$ \( T^{18} + 52 T^{16} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$11$ \( T^{18} + 89 T^{16} + \cdots + 32761 \) Copy content Toggle raw display
$13$ \( T^{18} \) Copy content Toggle raw display
$17$ \( (T^{9} + T^{8} + \cdots - 21064)^{2} \) Copy content Toggle raw display
$19$ \( T^{18} + 191 T^{16} + \cdots + 1827904 \) Copy content Toggle raw display
$23$ \( (T^{9} - 27 T^{8} + \cdots - 897001)^{2} \) Copy content Toggle raw display
$29$ \( (T^{9} + 14 T^{8} + \cdots + 710137)^{2} \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 9847094208064 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 147555415323121 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 61985065024 \) Copy content Toggle raw display
$43$ \( (T^{9} - 19 T^{8} + \cdots + 29107)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 2221554478144 \) Copy content Toggle raw display
$53$ \( (T^{9} + 30 T^{8} + \cdots - 2644993)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 52199931402304 \) Copy content Toggle raw display
$61$ \( (T^{9} + 12 T^{8} + \cdots + 40768)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 18\!\cdots\!89 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 59644873729 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 18\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( (T^{9} - T^{8} + \cdots - 22399)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 6173316667456 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 517490319424 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 73710078016 \) Copy content Toggle raw display
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