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Properties

Label 1520.3.h.b
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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Show commands: Magma / PariGP / SageMath

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: \( x^{12} + 62x^{10} + 1445x^{8} + 15924x^{6} + 83244x^{4} + 170640x^{2} + 55600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{9}\cdot 5 \)
Twist minimal: no (minimal twist has level 380)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{6} q^{5} + (\beta_{11} - \beta_{6} + 1) q^{7} + (\beta_{7} - \beta_{6} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{6} q^{5} + (\beta_{11} - \beta_{6} + 1) q^{7} + (\beta_{7} - \beta_{6} - 1) q^{9} + (\beta_{11} + \beta_{8} - \beta_{7} + 2) q^{11} + (\beta_{4} + \beta_1) q^{13} + \beta_{5} q^{15} + (\beta_{11} - \beta_{8} + \cdots - \beta_{6}) q^{17}+ \cdots + ( - \beta_{11} + 4 \beta_{10} + \cdots - 28) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{7} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{7} - 16 q^{9} + 32 q^{11} - 12 q^{17} - 24 q^{19} - 4 q^{23} + 60 q^{25} - 40 q^{35} - 124 q^{39} + 176 q^{43} - 40 q^{45} + 72 q^{47} - 24 q^{49} + 140 q^{57} + 152 q^{61} - 48 q^{63} - 148 q^{73} + 376 q^{77} - 468 q^{81} + 208 q^{83} + 84 q^{87} - 184 q^{93} - 392 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 62x^{10} + 1445x^{8} + 15924x^{6} + 83244x^{4} + 170640x^{2} + 55600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 59\nu^{11} + 5868\nu^{9} + 184075\nu^{7} + 2290126\nu^{5} + 10898236\nu^{3} + 13285720\nu ) / 407880 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -239\nu^{11} - 15993\nu^{9} - 373210\nu^{7} - 3571816\nu^{5} - 13082536\nu^{3} - 14157280\nu ) / 407880 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -401\nu^{11} - 20007\nu^{9} - 344590\nu^{7} - 2594164\nu^{5} - 8665084\nu^{3} - 10740520\nu ) / 203940 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -113\nu^{11} - 5661\nu^{9} - 95740\nu^{7} - 650992\nu^{5} - 1502992\nu^{3} - 417160\nu ) / 37080 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -113\nu^{10} - 5661\nu^{8} - 95740\nu^{6} - 650992\nu^{4} - 1502992\nu^{2} - 417160 ) / 37080 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -113\nu^{10} - 5661\nu^{8} - 95740\nu^{6} - 650992\nu^{4} - 1465912\nu^{2} - 46360 ) / 37080 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 301\nu^{10} + 14382\nu^{8} + 228185\nu^{6} + 1462904\nu^{4} + 3463424\nu^{2} + 1849460 ) / 67980 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -113\nu^{11} - 5661\nu^{9} - 95740\nu^{7} - 650992\nu^{5} - 1484452\nu^{3} - 139060\nu ) / 18540 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -361\nu^{10} - 17757\nu^{8} - 291230\nu^{6} - 1890134\nu^{4} - 4123544\nu^{2} - 1392200 ) / 67980 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 545\nu^{10} + 28107\nu^{8} + 495898\nu^{6} + 3578728\nu^{4} + 9148096\nu^{2} + 3524896 ) / 81576 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} - 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - 2\beta_{5} - 15\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{11} - 4\beta_{10} + \beta_{8} - 22\beta_{7} + 37\beta_{6} + 150 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -29\beta_{9} + 62\beta_{5} - 5\beta_{4} - 6\beta_{3} - 8\beta_{2} + 269\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -93\beta_{11} + 158\beta_{10} - 15\beta_{8} + 470\beta_{7} - 971\beta_{6} - 2674 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 721\beta_{9} - 1584\beta_{5} + 173\beta_{4} + 236\beta_{3} + 316\beta_{2} - 5403\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2413\beta_{11} - 4526\beta_{10} + 91\beta_{8} - 10310\beta_{7} + 23619\beta_{6} + 53398 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -17249\beta_{9} + 38364\beta_{5} - 4617\beta_{4} - 6848\beta_{3} - 9052\beta_{2} + 116231\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -59373\beta_{11} + 115918\beta_{10} + 2389\beta_{8} + 231734\beta_{7} - 560747\beta_{6} - 1144366 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 407025\beta_{9} - 910788\beta_{5} + 113529\beta_{4} + 177680\beta_{3} + 231836\beta_{2} - 2599023\beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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
− 4.83157i
− 3.53755i
− 3.42504i
− 3.14288i
− 2.03369i
− 0.630185i
0.630185i
2.03369i
3.14288i
3.42504i
3.53755i
4.83157i
0 − 4.83157i 0 2.23607 0 3.72856 0 −14.3441 0
721.2 0 − 3.53755i 0 −2.23607 0 −0.468611 0 −3.51429 0
721.3 0 − 3.42504i 0 2.23607 0 −9.18599 0 −2.73093 0
721.4 0 − 3.14288i 0 −2.23607 0 12.2192 0 −0.877687 0
721.5 0 − 2.03369i 0 −2.23607 0 −4.27843 0 4.86411 0
721.6 0 − 0.630185i 0 2.23607 0 3.98530 0 8.60287 0
721.7 0 0.630185i 0 2.23607 0 3.98530 0 8.60287 0
721.8 0 2.03369i 0 −2.23607 0 −4.27843 0 4.86411 0
721.9 0 3.14288i 0 −2.23607 0 12.2192 0 −0.877687 0
721.10 0 3.42504i 0 2.23607 0 −9.18599 0 −2.73093 0
721.11 0 3.53755i 0 −2.23607 0 −0.468611 0 −3.51429 0
721.12 0 4.83157i 0 2.23607 0 3.72856 0 −14.3441 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.b 12
4.b odd 2 1 380.3.e.a ✓ 12
12.b even 2 1 3420.3.o.a 12
19.b odd 2 1 inner 1520.3.h.b 12
20.d odd 2 1 1900.3.e.f 12
20.e even 4 2 1900.3.g.c 24
76.d even 2 1 380.3.e.a ✓ 12
228.b odd 2 1 3420.3.o.a 12
380.d even 2 1 1900.3.e.f 12
380.j odd 4 2 1900.3.g.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.3.e.a ✓ 12 4.b odd 2 1
380.3.e.a ✓ 12 76.d even 2 1
1520.3.h.b 12 1.a even 1 1 trivial
1520.3.h.b 12 19.b odd 2 1 inner
1900.3.e.f 12 20.d odd 2 1
1900.3.e.f 12 380.d even 2 1
1900.3.g.c 24 20.e even 4 2
1900.3.g.c 24 380.j odd 4 2
3420.3.o.a 12 12.b even 2 1
3420.3.o.a 12 228.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 62T_{3}^{10} + 1445T_{3}^{8} + 15924T_{3}^{6} + 83244T_{3}^{4} + 170640T_{3}^{2} + 55600 \) 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
$3$ \( T^{12} + 62 T^{10} + \cdots + 55600 \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} - 6 T^{5} + \cdots - 3344)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 16 T^{5} + \cdots - 43264)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 18127879600 \) Copy content Toggle raw display
$17$ \( (T^{6} + 6 T^{5} + \cdots + 818576)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 22\!\cdots\!61 \) Copy content Toggle raw display
$23$ \( (T^{6} + 2 T^{5} + \cdots - 752400)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 119374584217600 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} - 88 T^{5} + \cdots - 2451136)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 36 T^{5} + \cdots - 9167899456)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 492787956121600 \) Copy content Toggle raw display
$61$ \( (T^{6} - 76 T^{5} + \cdots - 1301500864)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{6} + 74 T^{5} + \cdots + 5637612816)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{6} - 104 T^{5} + \cdots - 2227337984)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
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