Properties

Label 1520.3.h.a
Level $1520$
Weight $3$
Character orbit 1520.h
Analytic conductor $41.417$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1520,3,Mod(721,1520)]
 
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([0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1520.721");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1520.h (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: \(41.4170001828\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 30x^{10} + 329x^{8} + 1620x^{6} + 3479x^{4} + 2470x^{2} + 55 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 95)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + \beta_{4} q^{5} + ( - \beta_{7} + \beta_{4} - 2) q^{7} + ( - \beta_{10} - \beta_{4} - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} + \beta_{4} q^{5} + ( - \beta_{7} + \beta_{4} - 2) q^{7} + ( - \beta_{10} - \beta_{4} - 4) q^{9} + ( - \beta_{10} + \beta_{7} + 2 \beta_{3} + \beta_{2} - 3) q^{11} + ( - \beta_{9} + \beta_1) q^{13} + \beta_{8} q^{15} + (\beta_{7} + 3 \beta_{4} + 2 \beta_{3} - \beta_{2} - 4) q^{17} + (\beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{19} + (2 \beta_{11} + \beta_{9} + \beta_{8} - 3 \beta_{6} - \beta_{5}) q^{21} + (\beta_{7} + 5 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2) q^{23} + 5 q^{25} + ( - \beta_{11} - 2 \beta_{8} - \beta_{6} - 2 \beta_{5} + 2 \beta_1) q^{27} + ( - \beta_{11} - \beta_{9} - 2 \beta_{8} + 4 \beta_{6} + 2 \beta_{5} + \beta_1) q^{29} + ( - \beta_{11} - \beta_{8} - 5 \beta_{6} - 5 \beta_{5} + 5 \beta_1) q^{31} + (2 \beta_{11} - 3 \beta_{9} - 4 \beta_{8} - 5 \beta_{6} + 3 \beta_1) q^{33} + (\beta_{10} + \beta_{7} - 2 \beta_{4} - \beta_{3} + \beta_{2} + 4) q^{35} + (3 \beta_{11} + \beta_{9} + 2 \beta_{8} - \beta_{6} - \beta_1) q^{37} + (\beta_{10} - 4 \beta_{7} - 5 \beta_{4} - 8 \beta_{3} - 5) q^{39} + ( - 5 \beta_{11} + 2 \beta_{9} + \beta_{8} + 3 \beta_{6} + \beta_{5} + 5 \beta_1) q^{41} + ( - 2 \beta_{10} + 2 \beta_{7} - 6 \beta_{4} + 3 \beta_{2} - 26) q^{43} + ( - 2 \beta_{10} + 2 \beta_{7} - 5 \beta_{4} + \beta_{3} + \beta_{2} - 3) q^{45} + ( - 4 \beta_{10} + 2 \beta_{7} - 8 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 6) q^{47} + (2 \beta_{10} - \beta_{7} - 15 \beta_{4} - 4 \beta_{3} - \beta_{2} + 7) q^{49} + ( - 3 \beta_{11} - 3 \beta_{9} + 2 \beta_{8} - 6 \beta_{6} + 11 \beta_1) q^{51} + (2 \beta_{11} - 6 \beta_{8} + 3 \beta_{6} - 8 \beta_{5}) q^{53} + ( - \beta_{10} - \beta_{7} - 4 \beta_{4} - 4 \beta_{3} - \beta_{2} + 1) q^{55} + ( - 5 \beta_{11} + 4 \beta_{10} + 4 \beta_{8} - 5 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} + \cdots + 6) q^{57}+ \cdots + (5 \beta_{10} - 5 \beta_{7} + 30 \beta_{4} - 14 \beta_{3} - \beta_{2} + 29) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 20 q^{7} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 20 q^{7} - 48 q^{9} - 32 q^{11} - 44 q^{17} - 8 q^{19} - 36 q^{23} + 60 q^{25} + 40 q^{35} - 76 q^{39} - 320 q^{43} - 40 q^{45} + 56 q^{47} + 72 q^{49} + 60 q^{57} - 296 q^{61} + 96 q^{63} - 244 q^{73} - 200 q^{77} - 372 q^{81} + 160 q^{83} + 160 q^{85} - 444 q^{87} + 296 q^{93} + 80 q^{95} + 312 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 30x^{10} + 329x^{8} + 1620x^{6} + 3479x^{4} + 2470x^{2} + 55 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} + 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -11\nu^{10} - 435\nu^{8} - 5828\nu^{6} - 30456\nu^{4} - 49397\nu^{2} - 1209 ) / 2672 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25\nu^{10} + 685\nu^{8} + 6444\nu^{6} + 24280\nu^{4} + 31863\nu^{2} + 6695 ) / 2672 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 25\nu^{11} + 685\nu^{9} + 6444\nu^{7} + 24280\nu^{5} + 34535\nu^{3} + 28071\nu ) / 2672 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -23\nu^{11} - 697\nu^{9} - 7692\nu^{7} - 37568\nu^{5} - 76929\nu^{3} - 46707\nu ) / 1336 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -49\nu^{10} - 1209\nu^{8} - 10172\nu^{6} - 35832\nu^{4} - 52191\nu^{2} - 17531 ) / 2672 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -81\nu^{11} - 2353\nu^{9} - 24940\nu^{7} - 119816\nu^{5} - 260991\nu^{3} - 206995\nu ) / 2672 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 16\nu^{11} + 405\nu^{9} + 3376\nu^{7} + 9928\nu^{5} + 2864\nu^{3} - 15321\nu ) / 668 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -89\nu^{10} - 2305\nu^{8} - 19948\nu^{6} - 63992\nu^{4} - 45991\nu^{2} + 27869 ) / 2672 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -19\nu^{11} - 554\nu^{9} - 5846\nu^{7} - 27404\nu^{5} - 55505\nu^{3} - 35716\nu ) / 334 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} - \beta_{8} - \beta_{6} + \beta_{5} - 8\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} + 4\beta_{4} + \beta_{3} - 6\beta_{2} + 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -17\beta_{11} - 2\beta_{9} + 15\beta_{8} + 21\beta_{6} - 11\beta_{5} + 74\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -30\beta_{10} - 10\beta_{7} - 144\beta_{4} - 40\beta_{3} + 133\beta_{2} - 740 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 230\beta_{11} + 40\beta_{9} - 200\beta_{8} - 300\beta_{6} + 96\beta_{5} - 739\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 188\beta_{10} + 112\beta_{7} + 1012\beta_{4} + 280\beta_{3} - 734\beta_{2} + 3705 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -2872\beta_{11} - 600\beta_{9} + 2536\beta_{8} + 3768\beta_{6} - 776\beta_{5} + 7733\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -4512\beta_{10} - 3560\beta_{7} - 25896\beta_{4} - 6976\beta_{3} + 16321\beta_{2} - 77778 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 34537\beta_{11} + 8072\beta_{9} - 31121\beta_{8} - 44929\beta_{6} + 6033\beta_{5} - 83340\beta_1 ) / 2 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
721.1
0.151673i
1.14149i
3.38208i
2.36559i
2.84623i
1.88109i
1.88109i
2.84623i
2.36559i
3.38208i
1.14149i
0.151673i
0 5.10387i 0 2.23607 0 6.35478 0 −17.0495 0
721.2 0 4.13699i 0 −2.23607 0 −7.54444 0 −8.11468 0
721.3 0 4.08851i 0 2.23607 0 −4.56923 0 −7.71590 0
721.4 0 3.55563i 0 −2.23607 0 −11.0785 0 −3.64250 0
721.5 0 2.18419i 0 −2.23607 0 9.15076 0 4.22932 0
721.6 0 0.840697i 0 2.23607 0 −2.31342 0 8.29323 0
721.7 0 0.840697i 0 2.23607 0 −2.31342 0 8.29323 0
721.8 0 2.18419i 0 −2.23607 0 9.15076 0 4.22932 0
721.9 0 3.55563i 0 −2.23607 0 −11.0785 0 −3.64250 0
721.10 0 4.08851i 0 2.23607 0 −4.56923 0 −7.71590 0
721.11 0 4.13699i 0 −2.23607 0 −7.54444 0 −8.11468 0
721.12 0 5.10387i 0 2.23607 0 6.35478 0 −17.0495 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 721.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1520.3.h.a 12
4.b odd 2 1 95.3.c.a 12
12.b even 2 1 855.3.e.a 12
19.b odd 2 1 inner 1520.3.h.a 12
20.d odd 2 1 475.3.c.g 12
20.e even 4 2 475.3.d.c 24
76.d even 2 1 95.3.c.a 12
228.b odd 2 1 855.3.e.a 12
380.d even 2 1 475.3.c.g 12
380.j odd 4 2 475.3.d.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.3.c.a 12 4.b odd 2 1
95.3.c.a 12 76.d even 2 1
475.3.c.g 12 20.d odd 2 1
475.3.c.g 12 380.d even 2 1
475.3.d.c 24 20.e even 4 2
475.3.d.c 24 380.j odd 4 2
855.3.e.a 12 12.b even 2 1
855.3.e.a 12 228.b odd 2 1
1520.3.h.a 12 1.a even 1 1 trivial
1520.3.h.a 12 19.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 78T_{3}^{10} + 2325T_{3}^{8} + 32996T_{3}^{6} + 222364T_{3}^{4} + 590960T_{3}^{2} + 317680 \) acting on \(S_{3}^{\mathrm{new}}(1520, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 78 T^{10} + 2325 T^{8} + \cdots + 317680 \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} + 10 T^{5} - 115 T^{4} + \cdots + 51376)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 16 T^{5} - 340 T^{4} + \cdots + 1609984)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 918 T^{10} + \cdots + 1391784557680 \) Copy content Toggle raw display
$17$ \( (T^{6} + 22 T^{5} - 495 T^{4} + \cdots + 1301904)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 8 T^{11} + \cdots + 22\!\cdots\!61 \) Copy content Toggle raw display
$23$ \( (T^{6} + 18 T^{5} - 1835 T^{4} + \cdots - 97445584)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 5306 T^{10} + \cdots + 63\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{12} + 8576 T^{10} + \cdots + 20\!\cdots\!80 \) Copy content Toggle raw display
$37$ \( T^{12} + 8420 T^{10} + \cdots + 19\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{12} + 16912 T^{10} + \cdots + 20\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( (T^{6} + 160 T^{5} + 8220 T^{4} + \cdots + 10954816)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 28 T^{5} - 5116 T^{4} + \cdots - 253813824)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 20038 T^{10} + \cdots + 41\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{12} + 31458 T^{10} + \cdots + 89\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( (T^{6} + 148 T^{5} + 5576 T^{4} + \cdots - 76754624)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 15694 T^{10} + \cdots + 86\!\cdots\!80 \) Copy content Toggle raw display
$71$ \( T^{12} + 32960 T^{10} + \cdots + 17\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( (T^{6} + 122 T^{5} - 8255 T^{4} + \cdots + 5877225616)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + 27240 T^{10} + \cdots + 11\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( (T^{6} - 80 T^{5} - 13648 T^{4} + \cdots + 5457804544)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 52504 T^{10} + \cdots + 52\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{12} + 21540 T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
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