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Properties

Label 7524.2.l.d
Level $7524$
Weight $2$
Character orbit 7524.l
Analytic conductor $60.079$
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] = [7524,2,Mod(2089,7524)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7524, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("7524.2089");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7524 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7524.l (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: \(60.0794424808\)
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: \( x^{16} + 74x^{14} + 1837x^{12} + 17366x^{10} + 57480x^{8} + 38306x^{6} + 8933x^{4} + 686x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 836)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + ( - \beta_{12} + \beta_{9}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} + ( - \beta_{12} + \beta_{9}) q^{7} + ( - \beta_{9} - \beta_{3}) q^{11} + \beta_{7} q^{13} + (\beta_{9} - \beta_1) q^{17} + \beta_{11} q^{19} + (2 \beta_{5} - \beta_{3} + 2 \beta_{2} - 2) q^{23} + (\beta_{3} - \beta_{2} - 2) q^{25} + (\beta_{11} - \beta_{10} + \beta_{7}) q^{29} + ( - \beta_{13} - \beta_{4}) q^{31} + (2 \beta_{12} - 2 \beta_{9} + \beta_1) q^{35} + ( - \beta_{8} + \beta_{4}) q^{37} + ( - \beta_{15} - \beta_{12} + \cdots + \beta_{6}) q^{41}+ \cdots + (2 \beta_{13} + 3 \beta_{8} - \beta_{4}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{5} - 24 q^{23} - 28 q^{25} - 4 q^{49} + 44 q^{77}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 74x^{14} + 1837x^{12} + 17366x^{10} + 57480x^{8} + 38306x^{6} + 8933x^{4} + 686x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 39673247 \nu^{15} - 2657158964 \nu^{13} - 52329108362 \nu^{11} - 182220770938 \nu^{9} + \cdots + 1189360698604 \nu ) / 45468377144 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 415325191 \nu^{14} - 30577267843 \nu^{12} - 751401634738 \nu^{10} - 6928331950236 \nu^{8} + \cdots + 11557506603 ) / 22734188572 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 422494599 \nu^{14} + 31201721687 \nu^{12} + 771472697802 \nu^{10} + 7221764167260 \nu^{8} + \cdots - 18203442579 ) / 22734188572 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 182584133 \nu^{15} - 13303886527 \nu^{13} - 319966306328 \nu^{11} + \cdots + 2493031965299 \nu ) / 68202565716 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 430080043 \nu^{14} + 31760394781 \nu^{12} + 785222350582 \nu^{10} + 7349462064338 \nu^{8} + \cdots + 72859017299 ) / 11367094286 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2135645363 \nu^{14} + 157618846967 \nu^{12} + 3892291037066 \nu^{10} + 36326180207588 \nu^{8} + \cdots + 461752071169 ) / 45468377144 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2219999202 \nu^{14} + 163759605995 \nu^{12} + 4039760882676 \nu^{10} + 37606057000886 \nu^{8} + \cdots - 103840397011 ) / 45468377144 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3852502201 \nu^{15} - 286331138447 \nu^{13} - 7168778346766 \nu^{11} + \cdots - 3700961449493 \nu ) / 136405131432 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3852502201 \nu^{15} + 286331138447 \nu^{13} + 7168778346766 \nu^{11} + \cdots + 3837366580925 \nu ) / 136405131432 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2882764817 \nu^{15} + 581308957 \nu^{14} + 213616034255 \nu^{13} + 43813296821 \nu^{12} + \cdots + 362653301343 ) / 90936754288 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2882764817 \nu^{15} - 1211449363 \nu^{14} + 213616034255 \nu^{13} - 89654080683 \nu^{12} + \cdots - 380045775697 ) / 90936754288 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15539418547 \nu^{15} + 1148283893612 \nu^{13} + 28425306990034 \nu^{11} + \cdots + 3074469431936 \nu ) / 136405131432 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 16644744785 \nu^{15} - 1228187919439 \nu^{13} - 30316677941066 \nu^{11} + \cdots - 1353112146097 \nu ) / 136405131432 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 8956174976 \nu^{15} - 661888011889 \nu^{13} - 16388364736688 \nu^{11} + \cdots - 3218400678349 \nu ) / 68202565716 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 6778403599 \nu^{15} - 500933619310 \nu^{13} - 12402680861454 \nu^{11} + \cdots - 2887343769086 \nu ) / 45468377144 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{9} + \beta_{8} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{6} - 2\beta_{5} + \beta_{2} - 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{15} + \beta_{14} + 7\beta_{12} - 27\beta_{9} - 25\beta_{8} + 4\beta_{4} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8 \beta_{15} + 8 \beta_{12} + 12 \beta_{11} + 4 \beta_{10} - 8 \beta_{9} - 8 \beta_{7} - 60 \beta_{6} + \cdots + 179 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 151 \beta_{15} - 42 \beta_{14} + 19 \beta_{13} - 242 \beta_{12} + 753 \beta_{9} + 704 \beta_{8} + \cdots - 105 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 314 \beta_{15} - 314 \beta_{12} - 462 \beta_{11} - 166 \beta_{10} + 314 \beta_{9} + 280 \beta_{7} + \cdots - 4766 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4752 \beta_{15} + 1255 \beta_{14} - 823 \beta_{13} + 7216 \beta_{12} - 21453 \beta_{9} + \cdots + 3202 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 9984 \beta_{15} + 9984 \beta_{12} + 14688 \beta_{11} + 5280 \beta_{10} - 9984 \beta_{9} - 8176 \beta_{7} + \cdots + 134161 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 142626 \beta_{15} - 34546 \beta_{14} + 27657 \beta_{13} - 208531 \beta_{12} + 615869 \beta_{9} + \cdots - 93075 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 300516 \beta_{15} - 300516 \beta_{12} - 445754 \beta_{11} - 155278 \beta_{10} + 300516 \beta_{9} + \cdots - 3845674 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4217421 \beta_{15} + 922159 \beta_{14} - 864040 \beta_{13} + 5976274 \beta_{12} - 17728365 \beta_{9} + \cdots + 2657829 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 8890636 \beta_{15} + 8890636 \beta_{12} + 13337244 \beta_{11} + 4444028 \beta_{10} - 8890636 \beta_{9} + \cdots + 110903189 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 124082194 \beta_{15} - 24197318 \beta_{14} + 26333226 \beta_{13} - 170843878 \beta_{12} + \cdots - 75385864 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 261499052 \beta_{15} - 261499052 \beta_{12} - 397081696 \beta_{11} - 125916408 \beta_{10} + \cdots - 3205009018 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 3644126606 \beta_{15} + 625983801 \beta_{14} - 795130480 \beta_{13} + 4879812821 \beta_{12} + \cdots + 2132406887 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7524\mathbb{Z}\right)^\times\).

\(n\) \(2377\) \(3763\) \(4105\) \(6689\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2089.1
− 5.34957i
0.533068i
5.34957i
− 0.533068i
− 0.341530i
2.29273i
0.341530i
− 2.29273i
− 0.127946i
− 5.40315i
0.127946i
5.40315i
− 0.610246i
− 3.18463i
0.610246i
3.18463i
0 0 0 −2.85168 0 − 3.70882i 0 0 0
2089.2 0 0 0 −2.85168 0 − 3.70882i 0 0 0
2089.3 0 0 0 −2.85168 0 3.70882i 0 0 0
2089.4 0 0 0 −2.85168 0 3.70882i 0 0 0
2089.5 0 0 0 −0.783036 0 − 3.52098i 0 0 0
2089.6 0 0 0 −0.783036 0 − 3.52098i 0 0 0
2089.7 0 0 0 −0.783036 0 3.52098i 0 0 0
2089.8 0 0 0 −0.783036 0 3.52098i 0 0 0
2089.9 0 0 0 0.691312 0 − 1.13040i 0 0 0
2089.10 0 0 0 0.691312 0 − 1.13040i 0 0 0
2089.11 0 0 0 0.691312 0 1.13040i 0 0 0
2089.12 0 0 0 0.691312 0 1.13040i 0 0 0
2089.13 0 0 0 1.94341 0 − 1.25280i 0 0 0
2089.14 0 0 0 1.94341 0 − 1.25280i 0 0 0
2089.15 0 0 0 1.94341 0 1.25280i 0 0 0
2089.16 0 0 0 1.94341 0 1.25280i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2089.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7524.2.l.d 16
3.b odd 2 1 836.2.b.b ✓ 16
11.b odd 2 1 inner 7524.2.l.d 16
19.b odd 2 1 inner 7524.2.l.d 16
33.d even 2 1 836.2.b.b ✓ 16
57.d even 2 1 836.2.b.b ✓ 16
209.d even 2 1 inner 7524.2.l.d 16
627.b odd 2 1 836.2.b.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
836.2.b.b ✓ 16 3.b odd 2 1
836.2.b.b ✓ 16 33.d even 2 1
836.2.b.b ✓ 16 57.d even 2 1
836.2.b.b ✓ 16 627.b odd 2 1
7524.2.l.d 16 1.a even 1 1 trivial
7524.2.l.d 16 11.b odd 2 1 inner
7524.2.l.d 16 19.b odd 2 1 inner
7524.2.l.d 16 209.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + T_{5}^{3} - 6T_{5}^{2} - T_{5} + 3 \) acting on \(S_{2}^{\mathrm{new}}(7524, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{4} + T^{3} - 6 T^{2} + \cdots + 3)^{4} \) Copy content Toggle raw display
$7$ \( (T^{8} + 29 T^{6} + \cdots + 342)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 8 T^{6} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 69 T^{6} + \cdots + 6574)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 68 T^{6} + \cdots + 608)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 16983563041 \) Copy content Toggle raw display
$23$ \( (T^{4} + 6 T^{3} + \cdots - 576)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} - 223 T^{6} + \cdots + 7159086)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 145 T^{6} + \cdots + 14013)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 86 T^{6} + \cdots + 24912)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 241 T^{6} + \cdots + 3786624)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 253 T^{6} + \cdots + 1368)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 58 T^{2} + \cdots - 72)^{4} \) Copy content Toggle raw display
$53$ \( (T^{8} + 224 T^{6} + \cdots + 3996992)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 302 T^{6} + \cdots + 2017872)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 204 T^{6} + \cdots + 87552)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 265 T^{6} + \cdots + 12332997)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 271 T^{6} + \cdots + 29237)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 500 T^{6} + \cdots + 133166592)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 228 T^{6} + \cdots + 105184)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 385 T^{6} + \cdots + 24502742)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 334 T^{6} + \cdots + 2768)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 418 T^{6} + \cdots + 27129168)^{2} \) Copy content Toggle raw display
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