Properties

Label 1932.4.a.j
Level $1932$
Weight $4$
Character orbit 1932.a
Self dual yes
Analytic conductor $113.992$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1932,4,Mod(1,1932)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1932, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1932.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1932.a (trivial)

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: yes
Analytic conductor: \(113.991690131\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3 x^{8} - 539 x^{7} + 1103 x^{6} + 87229 x^{5} - 136227 x^{4} - 4470966 x^{3} + 7400304 x^{2} + 54849888 x - 88928768 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + (\beta_1 - 2) q^{5} - 7 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + (\beta_1 - 2) q^{5} - 7 q^{7} + 9 q^{9} + (\beta_{2} - 2 \beta_1 + 6) q^{11} + (\beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} - 3) q^{13} + ( - 3 \beta_1 + 6) q^{15} + (\beta_{8} + \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 - 2) q^{17} + (\beta_{7} + \beta_{6} + 3 \beta_{5} - \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 4) q^{19} + 21 q^{21} + 23 q^{23} + (2 \beta_{7} + 3 \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} - 3 \beta_1 - 2) q^{25} - 27 q^{27} + (\beta_{8} - 3 \beta_{7} - \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + \cdots - 19) q^{29}+ \cdots + (9 \beta_{2} - 18 \beta_1 + 54) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 27 q^{3} - 15 q^{5} - 63 q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 27 q^{3} - 15 q^{5} - 63 q^{7} + 81 q^{9} + 49 q^{11} - 19 q^{13} + 45 q^{15} - 12 q^{17} + 63 q^{19} + 189 q^{21} + 207 q^{23} - 14 q^{25} - 243 q^{27} - 193 q^{29} - 400 q^{31} - 147 q^{33} + 105 q^{35} - 253 q^{37} + 57 q^{39} - 710 q^{41} - 161 q^{43} - 135 q^{45} + 292 q^{47} + 441 q^{49} + 36 q^{51} - 867 q^{53} - 1734 q^{55} - 189 q^{57} - 259 q^{59} - 1105 q^{61} - 567 q^{63} - q^{65} - 338 q^{67} - 621 q^{69} + 374 q^{71} + 6 q^{73} + 42 q^{75} - 343 q^{77} + 888 q^{79} + 729 q^{81} + 656 q^{83} - 1262 q^{85} + 579 q^{87} - 858 q^{89} + 133 q^{91} + 1200 q^{93} + 2832 q^{95} + 757 q^{97} + 441 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3 x^{8} - 539 x^{7} + 1103 x^{6} + 87229 x^{5} - 136227 x^{4} - 4470966 x^{3} + 7400304 x^{2} + 54849888 x - 88928768 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2250867111647 \nu^{8} + 25236895098535 \nu^{7} + \cdots - 17\!\cdots\!00 ) / 53\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2224445940137 \nu^{8} + 19994547038797 \nu^{7} + \cdots + 16\!\cdots\!60 ) / 35\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1288175780765 \nu^{8} + 2470982247669 \nu^{7} - 774925024303723 \nu^{6} + \cdots + 15\!\cdots\!88 ) / 17\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13114965317465 \nu^{8} + 63509766836729 \nu^{7} + \cdots + 82\!\cdots\!72 ) / 10\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5062710643517 \nu^{8} - 11958491988061 \nu^{7} + \cdots - 25\!\cdots\!72 ) / 35\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3063150821923 \nu^{8} + 6394399082866 \nu^{7} + \cdots + 58\!\cdots\!54 ) / 13\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 7857872197731 \nu^{8} + 24519686027943 \nu^{7} + \cdots - 10\!\cdots\!20 ) / 17\!\cdots\!24 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{7} + 3\beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 119 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -5\beta_{8} - \beta_{7} + 20\beta_{6} + 5\beta_{5} + 16\beta_{4} - \beta_{3} + 20\beta_{2} + 206\beta _1 + 96 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{8} + 556 \beta_{7} + 993 \beta_{6} + 132 \beta_{5} + 361 \beta_{4} - 295 \beta_{3} + 212 \beta_{2} + 665 \beta _1 + 25751 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 1864 \beta_{8} + 63 \beta_{7} + 8924 \beta_{6} + 2240 \beta_{5} + 5438 \beta_{4} + 628 \beta_{3} + 6550 \beta_{2} + 52289 \beta _1 + 64841 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2795 \beta_{8} + 150429 \beta_{7} + 315149 \beta_{6} + 72652 \beta_{5} + 107182 \beta_{4} - 83781 \beta_{3} + 56354 \beta_{2} + 287387 \beta _1 + 6565909 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 575859 \beta_{8} + 222445 \beta_{7} + 3345555 \beta_{6} + 918817 \beta_{5} + 1704284 \beta_{4} + 310292 \beta_{3} + 2017429 \beta_{2} + 14453682 \beta _1 + 28447446 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2213256 \beta_{8} + 41868634 \beta_{7} + 99089741 \beta_{6} + 28408746 \beta_{5} + 31957005 \beta_{4} - 23482965 \beta_{3} + 17222273 \beta_{2} + 108631875 \beta _1 + 1803057409 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−16.1826
−12.8419
−8.35038
−4.15941
1.61145
4.36390
7.36496
13.2771
17.9169
0 −3.00000 0 −18.1826 0 −7.00000 0 9.00000 0
1.2 0 −3.00000 0 −14.8419 0 −7.00000 0 9.00000 0
1.3 0 −3.00000 0 −10.3504 0 −7.00000 0 9.00000 0
1.4 0 −3.00000 0 −6.15941 0 −7.00000 0 9.00000 0
1.5 0 −3.00000 0 −0.388555 0 −7.00000 0 9.00000 0
1.6 0 −3.00000 0 2.36390 0 −7.00000 0 9.00000 0
1.7 0 −3.00000 0 5.36496 0 −7.00000 0 9.00000 0
1.8 0 −3.00000 0 11.2771 0 −7.00000 0 9.00000 0
1.9 0 −3.00000 0 15.9169 0 −7.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)
\(23\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1932.4.a.j 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1932.4.a.j 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{9} + 15 T_{5}^{8} - 443 T_{5}^{7} - 6107 T_{5}^{6} + 55861 T_{5}^{5} + 651995 T_{5}^{4} - 2196982 T_{5}^{3} - 15814876 T_{5}^{2} + 33388104 T_{5} + 15217536 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1932))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( (T + 3)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} + 15 T^{8} - 443 T^{7} + \cdots + 15217536 \) Copy content Toggle raw display
$7$ \( (T + 7)^{9} \) Copy content Toggle raw display
$11$ \( T^{9} - 49 T^{8} + \cdots - 6547555325952 \) Copy content Toggle raw display
$13$ \( T^{9} + 19 T^{8} + \cdots + 5749606538928 \) Copy content Toggle raw display
$17$ \( T^{9} + 12 T^{8} + \cdots + 35\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{9} - 63 T^{8} + \cdots + 29\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T - 23)^{9} \) Copy content Toggle raw display
$29$ \( T^{9} + 193 T^{8} + \cdots + 40\!\cdots\!28 \) Copy content Toggle raw display
$31$ \( T^{9} + 400 T^{8} + \cdots + 76\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{9} + 253 T^{8} + \cdots + 47\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( T^{9} + 710 T^{8} + \cdots - 20\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( T^{9} + 161 T^{8} + \cdots - 14\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( T^{9} - 292 T^{8} + \cdots + 81\!\cdots\!12 \) Copy content Toggle raw display
$53$ \( T^{9} + 867 T^{8} + \cdots + 11\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{9} + 259 T^{8} + \cdots + 15\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{9} + 1105 T^{8} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{9} + 338 T^{8} + \cdots + 17\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T^{9} - 374 T^{8} + \cdots - 13\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{9} - 6 T^{8} + \cdots + 64\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{9} - 888 T^{8} + \cdots - 44\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{9} - 656 T^{8} + \cdots + 76\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{9} + 858 T^{8} + \cdots + 35\!\cdots\!28 \) Copy content Toggle raw display
$97$ \( T^{9} - 757 T^{8} + \cdots - 30\!\cdots\!48 \) Copy content Toggle raw display
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