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Properties

Label 1520.2.g.g
Level $1520$
Weight $2$
Character orbit 1520.g
Analytic conductor $12.137$
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] = [1520,2,Mod(1519,1520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1520.1519");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.g (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: \(12.1372611072\)
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: \( x^{16} - 20x^{14} + 271x^{12} - 2000x^{10} + 10645x^{8} - 29570x^{6} + 58816x^{4} - 56840x^{2} + 38416 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12} \)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + \beta_{15} q^{5} + \beta_{10} q^{7} + (\beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} + \beta_{15} q^{5} + \beta_{10} q^{7} + (\beta_1 - 2) q^{9} + ( - \beta_{4} + \beta_{3}) q^{11} + ( - \beta_{10} + \beta_{8}) q^{13} + (\beta_{14} - \beta_{12} + \beta_{6} + \cdots + 2) q^{15}+ \cdots + (\beta_{9} + \beta_{7} + \cdots + 4 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{9} + 32 q^{15} - 8 q^{19} - 8 q^{25} + 96 q^{31} - 24 q^{45} - 8 q^{49} - 24 q^{51} - 72 q^{59} - 24 q^{61} - 96 q^{71} - 24 q^{75} + 32 q^{79} - 8 q^{81} - 56 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 20x^{14} + 271x^{12} - 2000x^{10} + 10645x^{8} - 29570x^{6} + 58816x^{4} - 56840x^{2} + 38416 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4429 \nu^{14} - 77314 \nu^{12} + 1007855 \nu^{10} - 6294330 \nu^{8} + 31135885 \nu^{6} + \cdots + 469265976 ) / 125016960 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 397497 \nu^{14} - 7515898 \nu^{12} + 100144915 \nu^{10} - 696224210 \nu^{8} + \cdots - 12772202408 ) / 6125831040 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2445217 \nu^{14} + 50395018 \nu^{12} - 682415115 \nu^{10} + 5182162850 \nu^{8} + \cdots + 81350328808 ) / 18377493120 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 32348 \nu^{14} + 671313 \nu^{12} - 9132730 \nu^{10} + 68382760 \nu^{8} - 357807455 \nu^{6} + \cdots + 1133374508 ) / 191432220 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 258 \nu^{14} - 4537 \nu^{12} + 58710 \nu^{10} - 366660 \nu^{8} + 1709600 \nu^{6} - 2238960 \nu^{4} + \cdots + 3806333 ) / 976695 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 397497 \nu^{15} - 7515898 \nu^{13} + 100144915 \nu^{11} - 696224210 \nu^{9} + \cdots - 6646371368 \nu ) / 12251662080 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8087003 \nu^{15} + 15441888 \nu^{14} + 174764750 \nu^{13} - 258490848 \nu^{12} + \cdots - 419116013568 ) / 257284903680 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3060399 \nu^{15} + 28801046 \nu^{13} - 243837925 \nu^{11} - 1681047170 \nu^{9} + \cdots - 610375641064 \nu ) / 85761634560 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8087003 \nu^{15} + 15441888 \nu^{14} - 174764750 \nu^{13} - 258490848 \nu^{12} + \cdots - 419116013568 ) / 257284903680 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 202691 \nu^{15} - 2601950 \nu^{13} + 26162145 \nu^{11} - 34220230 \nu^{9} + \cdots + 21347294920 \nu ) / 5593150080 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 8503883 \nu^{15} + 139049390 \nu^{13} - 1713925305 \nu^{11} + 9317827030 \nu^{9} + \cdots + 17822113400 \nu ) / 128642451840 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 794819 \nu^{15} - 14201470 \nu^{13} + 189226225 \nu^{11} - 1281113910 \nu^{9} + \cdots - 12558478520 \nu ) / 10720204320 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 12433839 \nu^{15} + 245478550 \nu^{13} - 3270857125 \nu^{11} + 23498819390 \nu^{9} + \cdots + 217078731800 \nu ) / 85761634560 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 6388160 \nu^{15} - 4527943 \nu^{14} - 125036448 \nu^{13} + 81780118 \nu^{12} + \cdots - 16824899112 ) / 42880817280 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 6388160 \nu^{15} + 4527943 \nu^{14} - 125036448 \nu^{13} - 81780118 \nu^{12} + \cdots + 16824899112 ) / 42880817280 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} - \beta_{7} + \beta_{4} + 5\beta_{2} - \beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} + 7\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -10\beta_{9} - 10\beta_{7} - \beta_{5} + 10\beta_{4} - 3\beta_{3} + 37\beta_{2} + 10\beta _1 - 35 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 10 \beta_{15} - 10 \beta_{14} - 12 \beta_{13} - 9 \beta_{12} + 100 \beta_{11} + 23 \beta_{10} + \cdots + 56 \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{15} - 4\beta_{14} - 15\beta_{5} + 88\beta _1 - 275 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 92 \beta_{15} + 92 \beta_{14} + 118 \beta_{13} + 65 \beta_{12} + 878 \beta_{11} + 237 \beta_{10} + \cdots - 458 \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 80 \beta_{15} - 80 \beta_{14} + 680 \beta_{9} + 680 \beta_{7} - 171 \beta_{5} - 920 \beta_{4} + \cdots - 2239 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 840\beta_{15} + 840\beta_{14} + 1102\beta_{13} + 429\beta_{12} - 3770\beta_{6} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1084 \beta_{15} + 1084 \beta_{14} + 5468 \beta_{9} + 5468 \beta_{7} + 1775 \beta_{5} - 8720 \beta_{4} + \cdots + 18475 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 7636 \beta_{15} + 7636 \beta_{14} + 10102 \beta_{13} + 2609 \beta_{12} - 66662 \beta_{11} + \cdots - 31186 \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -12520\beta_{15} + 12520\beta_{14} + 17595\beta_{5} - 56560\beta _1 + 153559 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 69080 \beta_{15} - 69080 \beta_{14} - 91750 \beta_{13} - 13925 \beta_{12} - 580894 \beta_{11} + \cdots + 259234 \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 132980 \beta_{15} + 132980 \beta_{14} - 356164 \beta_{9} - 356164 \beta_{7} + 169575 \beta_{5} + \cdots + 1283315 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -622124\beta_{15} - 622124\beta_{14} - 828294\beta_{13} - 53609\beta_{12} + 2165218\beta_{6} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1519.1
2.56586 − 1.48140i
−2.56586 − 1.48140i
−2.38641 − 1.37780i
2.38641 − 1.37780i
1.34352 − 0.775681i
−1.34352 − 0.775681i
−0.957255 − 0.552672i
0.957255 − 0.552672i
−0.957255 + 0.552672i
0.957255 + 0.552672i
1.34352 + 0.775681i
−1.34352 + 0.775681i
−2.38641 + 1.37780i
2.38641 + 1.37780i
2.56586 + 1.48140i
−2.56586 + 1.48140i
0 − 2.96281i 0 −0.602114 + 2.15348i 0 −0.562696 0 −5.77822 0
1519.2 0 − 2.96281i 0 −0.602114 + 2.15348i 0 0.562696 0 −5.77822 0
1519.3 0 − 2.75559i 0 2.03407 + 0.928731i 0 −4.29083 0 −4.59328 0
1519.4 0 − 2.75559i 0 2.03407 + 0.928731i 0 4.29083 0 −4.59328 0
1519.5 0 − 1.55136i 0 −2.03407 + 0.928731i 0 −1.46240 0 0.593276 0
1519.6 0 − 1.55136i 0 −2.03407 + 0.928731i 0 1.46240 0 0.593276 0
1519.7 0 − 1.10534i 0 0.602114 − 2.15348i 0 −2.26573 0 1.77822 0
1519.8 0 − 1.10534i 0 0.602114 − 2.15348i 0 2.26573 0 1.77822 0
1519.9 0 1.10534i 0 0.602114 + 2.15348i 0 −2.26573 0 1.77822 0
1519.10 0 1.10534i 0 0.602114 + 2.15348i 0 2.26573 0 1.77822 0
1519.11 0 1.55136i 0 −2.03407 − 0.928731i 0 −1.46240 0 0.593276 0
1519.12 0 1.55136i 0 −2.03407 − 0.928731i 0 1.46240 0 0.593276 0
1519.13 0 2.75559i 0 2.03407 − 0.928731i 0 −4.29083 0 −4.59328 0
1519.14 0 2.75559i 0 2.03407 − 0.928731i 0 4.29083 0 −4.59328 0
1519.15 0 2.96281i 0 −0.602114 − 2.15348i 0 −0.562696 0 −5.77822 0
1519.16 0 2.96281i 0 −0.602114 − 2.15348i 0 0.562696 0 −5.77822 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1519.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
76.d even 2 1 inner
380.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1520.2.g.g yes 16
4.b odd 2 1 1520.2.g.e ✓ 16
5.b even 2 1 inner 1520.2.g.g yes 16
19.b odd 2 1 1520.2.g.e ✓ 16
20.d odd 2 1 1520.2.g.e ✓ 16
76.d even 2 1 inner 1520.2.g.g yes 16
95.d odd 2 1 1520.2.g.e ✓ 16
380.d even 2 1 inner 1520.2.g.g yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1520.2.g.e ✓ 16 4.b odd 2 1
1520.2.g.e ✓ 16 19.b odd 2 1
1520.2.g.e ✓ 16 20.d odd 2 1
1520.2.g.e ✓ 16 95.d odd 2 1
1520.2.g.g yes 16 1.a even 1 1 trivial
1520.2.g.g yes 16 5.b even 2 1 inner
1520.2.g.g yes 16 76.d even 2 1 inner
1520.2.g.g yes 16 380.d 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 \(S_{2}^{\mathrm{new}}(1520, [\chi])\):

\( T_{3}^{8} + 20T_{3}^{6} + 129T_{3}^{4} + 290T_{3}^{2} + 196 \) Copy content Toggle raw display
\( T_{7}^{8} - 26T_{7}^{6} + 153T_{7}^{4} - 248T_{7}^{2} + 64 \) Copy content Toggle raw display
\( T_{31}^{4} - 24T_{31}^{3} + 198T_{31}^{2} - 648T_{31} + 672 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 20 T^{6} + \cdots + 196)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + 2 T^{6} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 26 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 40 T^{6} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 56 T^{6} + \cdots + 196)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 21 T^{2} + 96)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 2 T^{3} + \cdots + 361)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} - 90 T^{6} + \cdots + 28224)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 106 T^{6} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 24 T^{3} + \cdots + 672)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} - 110 T^{6} + \cdots + 50176)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 292 T^{6} + \cdots + 18113536)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 116 T^{6} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 240 T^{6} + \cdots + 4064256)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 288 T^{6} + \cdots + 13972644)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 18 T^{3} + \cdots - 2352)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 6 T^{3} + \cdots - 504)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 300 T^{6} + \cdots + 86436)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 24 T^{3} + \cdots + 672)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 234 T^{6} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} + \cdots + 448)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 216 T^{6} + \cdots + 589824)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 460 T^{6} + \cdots + 802816)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 230 T^{6} + \cdots + 3136)^{2} \) Copy content Toggle raw display
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