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Properties

Label 460.3.d.c
Level $460$
Weight $3$
Character orbit 460.d
Analytic conductor $12.534$
Analytic rank $0$
Dimension $20$
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] = [460,3,Mod(229,460)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(460, 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("460.229");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 460.d (of order \(2\), degree \(1\), 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: \(12.5340921606\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 62 x^{18} + 2901 x^{16} + 78832 x^{14} + 1920634 x^{12} + 40758500 x^{10} + \cdots + 95367431640625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + \beta_1 q^{5} - \beta_{16} q^{7} + (\beta_{11} - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + \beta_1 q^{5} - \beta_{16} q^{7} + (\beta_{11} - 4) q^{9} - \beta_{6} q^{11} + (\beta_{8} + \beta_{4}) q^{13} + ( - \beta_{16} + \beta_{14} + \cdots + \beta_1) q^{15}+ \cdots + (\beta_{19} + 2 \beta_{18} + \cdots - 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 72 q^{9} - 124 q^{25} - 128 q^{29} - 68 q^{31} - 100 q^{35} + 216 q^{39} - 68 q^{41} + 24 q^{49} + 28 q^{55} - 360 q^{59} + 56 q^{69} + 76 q^{71} - 52 q^{75} - 476 q^{81} + 36 q^{85} - 248 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 62 x^{18} + 2901 x^{16} + 78832 x^{14} + 1920634 x^{12} + 40758500 x^{10} + \cdots + 95367431640625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 104332 \nu^{18} - 18978041 \nu^{16} - 1050179868 \nu^{14} - 46528693276 \nu^{12} + \cdots - 59\!\cdots\!25 ) / 60\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 398004 \nu^{18} - 578127 \nu^{16} + 198994604 \nu^{14} - 11137881172 \nu^{12} + \cdots - 10\!\cdots\!75 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 50503979 \nu^{18} - 2067335448 \nu^{16} - 83558529954 \nu^{14} - 1430601261278 \nu^{12} + \cdots + 10\!\cdots\!50 ) / 86\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{19} + 62 \nu^{17} + 2901 \nu^{15} + 78832 \nu^{13} + 1920634 \nu^{11} + \cdots + 9460449218750 \nu ) / 3814697265625 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6020769 \nu^{19} + 618043303 \nu^{17} + 35946177744 \nu^{15} + 955254001808 \nu^{13} + \cdots + 83\!\cdots\!75 \nu ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 856804177 \nu^{18} + 276474156651 \nu^{16} + 11906576160648 \nu^{14} + \cdots + 46\!\cdots\!75 ) / 68\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2387757131 \nu^{18} + 168579714753 \nu^{16} + 7438611742344 \nu^{14} + \cdots + 40\!\cdots\!25 ) / 68\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1607659223 \nu^{19} - 86900244951 \nu^{17} - 3858535899048 \nu^{15} + \cdots - 77\!\cdots\!75 \nu ) / 86\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 101154573 \nu^{18} + 661789401 \nu^{16} + 15283508648 \nu^{14} - 2214894442264 \nu^{12} + \cdots - 52\!\cdots\!75 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 17679748 \nu^{18} - 1013392501 \nu^{16} - 36621613948 \nu^{14} - 827771001836 \nu^{12} + \cdots - 44\!\cdots\!25 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 18475756 \nu^{18} - 1012236247 \nu^{16} - 37019603156 \nu^{14} - 805495239492 \nu^{12} + \cdots - 41\!\cdots\!75 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 388866329 \nu^{19} + 31679391773 \nu^{17} + 1208693607304 \nu^{15} + \cdots + 19\!\cdots\!25 \nu ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1928784857 \nu^{19} + 143555220509 \nu^{17} + 5492439317032 \nu^{15} + \cdots + 91\!\cdots\!25 \nu ) / 57\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2170187549 \nu^{19} - 282151844913 \nu^{17} - 10913170104024 \nu^{15} + \cdots - 30\!\cdots\!25 \nu ) / 57\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 518818679 \nu^{19} + 61554556723 \nu^{17} + 2345456736904 \nu^{15} + \cdots + 60\!\cdots\!75 \nu ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 37064924 \nu^{18} - 2055797163 \nu^{16} - 73337669524 \nu^{14} - 1601244001268 \nu^{12} + \cdots - 78\!\cdots\!75 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 717869649 \nu^{19} - 29139543863 \nu^{17} - 998468218624 \nu^{15} + \cdots + 36\!\cdots\!25 \nu ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 595509149 \nu^{19} + 27396870363 \nu^{17} + 983977163124 \nu^{15} + \cdots + 57\!\cdots\!75 \nu ) / 11\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} - \beta_{11} + \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + \beta_{18} - 9\beta_{16} - 7\beta_{15} + 6\beta_{13} + 4\beta_{9} - 3\beta_{6} + 6\beta_{5} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 25 \beta_{17} - 33 \beta_{12} - 21 \beta_{11} + 12 \beta_{10} + 2 \beta_{8} + 2 \beta_{7} + 102 \beta_{4} + \cdots - 204 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11 \beta_{19} + 3 \beta_{18} + 59 \beta_{16} + 5 \beta_{15} + 56 \beta_{14} - 262 \beta_{13} + \cdots - 270 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 571 \beta_{17} + 360 \beta_{12} + 866 \beta_{11} - 124 \beta_{10} - 472 \beta_{8} + 352 \beta_{7} + \cdots + 6202 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 632 \beta_{19} + 600 \beta_{18} + 6472 \beta_{16} + 6264 \beta_{15} - 4344 \beta_{14} + \cdots + 2908 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 18016 \beta_{17} + 15064 \beta_{12} + 24104 \beta_{11} + 8208 \beta_{10} + 11620 \beta_{8} + \cdots - 112095 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3112 \beta_{19} + 2360 \beta_{18} - 42520 \beta_{16} - 11304 \beta_{15} + 147352 \beta_{14} + \cdots - 176119 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 248288 \beta_{17} - 743131 \beta_{12} + 123767 \beta_{11} - 214956 \beta_{10} - 89832 \beta_{8} + \cdots - 5182089 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 757899 \beta_{19} - 866675 \beta_{18} + 985211 \beta_{16} - 414363 \beta_{15} - 1960472 \beta_{14} + \cdots - 5424563 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 5962843 \beta_{17} + 11703235 \beta_{12} + 1835919 \beta_{11} - 1674076 \beta_{10} - 2173850 \beta_{8} + \cdots + 26614760 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 10169727 \beta_{19} + 10677263 \beta_{18} + 55247703 \beta_{16} + 54964217 \beta_{15} + \cdots + 59247974 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 18072729 \beta_{17} - 31831152 \beta_{12} + 70404758 \beta_{11} + 71571128 \beta_{10} + \cdots + 2761850666 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 336222704 \beta_{19} - 174096176 \beta_{18} - 2089999680 \beta_{16} - 957014464 \beta_{15} + \cdots + 3461235336 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 6159451248 \beta_{17} - 3054261520 \beta_{12} - 12566944720 \beta_{11} - 4821081440 \beta_{10} + \cdots - 49127524399 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 6586828528 \beta_{19} - 9138272816 \beta_{18} - 29231402240 \beta_{16} - 48772751872 \beta_{15} + \cdots - 18668075919 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 56177254704 \beta_{17} + 48212732601 \beta_{12} - 169423457329 \beta_{11} + 80880135288 \beta_{10} + \cdots + 1290808700403 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 358918921017 \beta_{19} + 277420383177 \beta_{18} + 149442562015 \beta_{16} + 8356983601 \beta_{15} + \cdots + 1530553950941 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/460\mathbb{Z}\right)^\times\).

\(n\) \(231\) \(277\) \(281\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
229.1
−4.65848 − 1.81620i
4.65848 + 1.81620i
−2.61634 + 4.26084i
2.61634 − 4.26084i
−3.05211 − 3.96038i
3.05211 + 3.96038i
−2.71432 + 4.19910i
2.71432 − 4.19910i
−1.33057 + 4.81971i
1.33057 − 4.81971i
−1.33057 − 4.81971i
1.33057 + 4.81971i
−2.71432 − 4.19910i
2.71432 + 4.19910i
−3.05211 + 3.96038i
3.05211 − 3.96038i
−2.61634 − 4.26084i
2.61634 + 4.26084i
−4.65848 + 1.81620i
4.65848 − 1.81620i
0 − 5.07205i 0 −4.65848 − 1.81620i 0 −3.07132 0 −16.7257 0
229.2 0 − 5.07205i 0 4.65848 + 1.81620i 0 3.07132 0 −16.7257 0
229.3 0 − 4.41085i 0 −2.61634 + 4.26084i 0 12.7035 0 −10.4556 0
229.4 0 − 4.41085i 0 2.61634 − 4.26084i 0 −12.7035 0 −10.4556 0
229.5 0 − 3.08989i 0 −3.05211 − 3.96038i 0 6.21474 0 −0.547411 0
229.6 0 − 3.08989i 0 3.05211 + 3.96038i 0 −6.21474 0 −0.547411 0
229.7 0 − 2.55211i 0 −2.71432 + 4.19910i 0 −1.70335 0 2.48674 0
229.8 0 − 2.55211i 0 2.71432 − 4.19910i 0 1.70335 0 2.48674 0
229.9 0 − 1.32589i 0 −1.33057 + 4.81971i 0 −6.21806 0 7.24200 0
229.10 0 − 1.32589i 0 1.33057 − 4.81971i 0 6.21806 0 7.24200 0
229.11 0 1.32589i 0 −1.33057 − 4.81971i 0 −6.21806 0 7.24200 0
229.12 0 1.32589i 0 1.33057 + 4.81971i 0 6.21806 0 7.24200 0
229.13 0 2.55211i 0 −2.71432 − 4.19910i 0 −1.70335 0 2.48674 0
229.14 0 2.55211i 0 2.71432 + 4.19910i 0 1.70335 0 2.48674 0
229.15 0 3.08989i 0 −3.05211 + 3.96038i 0 6.21474 0 −0.547411 0
229.16 0 3.08989i 0 3.05211 − 3.96038i 0 −6.21474 0 −0.547411 0
229.17 0 4.41085i 0 −2.61634 − 4.26084i 0 12.7035 0 −10.4556 0
229.18 0 4.41085i 0 2.61634 + 4.26084i 0 −12.7035 0 −10.4556 0
229.19 0 5.07205i 0 −4.65848 + 1.81620i 0 −3.07132 0 −16.7257 0
229.20 0 5.07205i 0 4.65848 − 1.81620i 0 3.07132 0 −16.7257 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 229.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
23.b odd 2 1 inner
115.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.3.d.c ✓ 20
5.b even 2 1 inner 460.3.d.c ✓ 20
5.c odd 4 2 2300.3.f.f 20
23.b odd 2 1 inner 460.3.d.c ✓ 20
115.c odd 2 1 inner 460.3.d.c ✓ 20
115.e even 4 2 2300.3.f.f 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.3.d.c ✓ 20 1.a even 1 1 trivial
460.3.d.c ✓ 20 5.b even 2 1 inner
460.3.d.c ✓ 20 23.b odd 2 1 inner
460.3.d.c ✓ 20 115.c odd 2 1 inner
2300.3.f.f 20 5.c odd 4 2
2300.3.f.f 20 115.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(460, [\chi])\):

\( T_{3}^{10} + 63T_{3}^{8} + 1396T_{3}^{6} + 13113T_{3}^{4} + 50195T_{3}^{2} + 54716 \) Copy content Toggle raw display
\( T_{7}^{10} - 251T_{7}^{8} + 16937T_{7}^{6} - 419784T_{7}^{4} + 3354724T_{7}^{2} - 6595712 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{10} + 63 T^{8} + \cdots + 54716)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 95367431640625 \) Copy content Toggle raw display
$7$ \( (T^{10} - 251 T^{8} + \cdots - 6595712)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + 749 T^{8} + \cdots + 2364716088)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + 1191 T^{8} + \cdots + 2769997500)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} - 981 T^{8} + \cdots - 12903200)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 14941760470200)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 17\!\cdots\!01 \) Copy content Toggle raw display
$29$ \( (T^{5} + 32 T^{4} + \cdots - 643000)^{4} \) Copy content Toggle raw display
$31$ \( (T^{5} + 17 T^{4} + \cdots + 3731548)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots - 8760963123200)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} + 17 T^{4} + \cdots - 63339680)^{4} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 668852527232)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 10\!\cdots\!36)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 50\!\cdots\!52)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} + 90 T^{4} + \cdots + 1885410080)^{4} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 290658217164408)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots - 12\!\cdots\!72)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} - 19 T^{4} + \cdots - 110070650)^{4} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 27\!\cdots\!72)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 487502560702592)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 90\!\cdots\!08)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots - 31\!\cdots\!68)^{2} \) Copy content Toggle raw display
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