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Properties

Label 1856.4.a.bf
Level $1856$
Weight $4$
Character orbit 1856.a
Self dual yes
Analytic conductor $109.508$
Analytic rank $1$
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] = [1856,4,Mod(1,1856)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1856, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1856.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.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: \(109.507544971\)
Analytic rank: \(1\)
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: \( x^{9} - 4x^{8} - 138x^{7} + 394x^{6} + 5872x^{5} - 10822x^{4} - 85158x^{3} + 30654x^{2} + 439999x + 396802 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 928)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{4} - \beta_1 - 1) q^{5} + (\beta_{6} + 1) q^{7} + (\beta_{8} + \beta_{5} - \beta_{4} + \cdots + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{4} - \beta_1 - 1) q^{5} + (\beta_{6} + 1) q^{7} + (\beta_{8} + \beta_{5} - \beta_{4} + \cdots + 5) q^{9}+ \cdots + ( - 12 \beta_{8} - 2 \beta_{7} + \cdots - 469) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 4 q^{3} - 10 q^{5} + 12 q^{7} + 49 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 4 q^{3} - 10 q^{5} + 12 q^{7} + 49 q^{9} - 64 q^{11} - 70 q^{13} + 170 q^{15} - 66 q^{17} - 42 q^{19} - 76 q^{21} + 40 q^{23} + 111 q^{25} - 322 q^{27} + 261 q^{29} - 64 q^{31} - 52 q^{33} - 496 q^{35} + 54 q^{37} + 590 q^{39} - 378 q^{41} + 32 q^{43} - 1046 q^{45} + 1164 q^{47} - 351 q^{49} - 376 q^{51} - 278 q^{53} + 614 q^{55} + 28 q^{57} - 640 q^{59} - 1054 q^{61} + 1660 q^{63} - 708 q^{65} - 1184 q^{67} - 188 q^{69} + 1988 q^{71} - 750 q^{73} - 3126 q^{75} - 1260 q^{77} + 2916 q^{79} + 293 q^{81} - 2832 q^{83} - 56 q^{85} - 116 q^{87} - 370 q^{89} - 3016 q^{91} + 1696 q^{93} + 4412 q^{95} - 2234 q^{97} - 4118 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 4x^{8} - 138x^{7} + 394x^{6} + 5872x^{5} - 10822x^{4} - 85158x^{3} + 30654x^{2} + 439999x + 396802 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2129 \nu^{8} - 11757 \nu^{7} - 264339 \nu^{6} + 1175057 \nu^{5} + 9556105 \nu^{4} + \cdots + 369231982 ) / 4740120 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 39 \nu^{8} + 317 \nu^{7} + 4409 \nu^{6} - 34357 \nu^{5} - 127235 \nu^{4} + 1014083 \nu^{3} + \cdots - 860902 ) / 83160 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 347 \nu^{8} - 2165 \nu^{7} - 43679 \nu^{6} + 235173 \nu^{5} + 1598859 \nu^{4} - 7355187 \nu^{3} + \cdots + 78704822 ) / 316008 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3196 \nu^{8} + 17103 \nu^{7} + 412971 \nu^{6} - 1812133 \nu^{5} - 15686240 \nu^{4} + \cdots - 787744538 ) / 2370060 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 8339 \nu^{8} + 49617 \nu^{7} + 1041729 \nu^{6} - 5309147 \nu^{5} - 37053595 \nu^{4} + \cdots - 1519251202 ) / 4740120 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1333 \nu^{8} - 6999 \nu^{7} - 171603 \nu^{6} + 734299 \nu^{5} + 6418745 \nu^{4} - 22288601 \nu^{3} + \cdots + 259520594 ) / 677160 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11597 \nu^{8} - 66681 \nu^{7} - 1481127 \nu^{6} + 7151861 \nu^{5} + 55355365 \nu^{4} + \cdots + 2604377566 ) / 4740120 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{5} - \beta_{4} + 2\beta _1 + 32 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{8} - 2\beta_{7} - \beta_{5} - 7\beta_{4} - 2\beta_{3} - \beta_{2} + 64\beta _1 + 37 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 96\beta_{8} - 17\beta_{7} - 6\beta_{6} + 73\beta_{5} - 105\beta_{4} + \beta_{3} + 5\beta_{2} + 247\beta _1 + 1882 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 580 \beta_{8} - 258 \beta_{7} - 96 \beta_{6} + 88 \beta_{5} - 880 \beta_{4} - 114 \beta_{3} + \cdots + 5092 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8663 \beta_{8} - 2414 \beta_{7} - 1284 \beta_{6} + 5625 \beta_{5} - 10253 \beta_{4} + 250 \beta_{3} + \cdots + 134664 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 64308 \beta_{8} - 26950 \beta_{7} - 14772 \beta_{6} + 19853 \beta_{5} - 92853 \beta_{4} - 5062 \beta_{3} + \cdots + 611739 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 788310 \beta_{8} - 261219 \beta_{7} - 161082 \beta_{6} + 461973 \beta_{5} - 984441 \beta_{4} + \cdots + 10714546 \) Copy content Toggle raw display

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

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
9.74094
6.50540
4.87969
3.39022
−1.51339
−1.95108
−2.38920
−7.02483
−7.63776
0 −9.74094 0 −17.5483 0 10.7391 0 67.8859 0
1.2 0 −6.50540 0 1.74434 0 26.0581 0 15.3202 0
1.3 0 −4.87969 0 −10.4780 0 −23.3159 0 −3.18861 0
1.4 0 −3.39022 0 19.3767 0 −7.92881 0 −15.5064 0
1.5 0 1.51339 0 6.33195 0 1.77839 0 −24.7097 0
1.6 0 1.95108 0 −18.0628 0 22.0180 0 −23.1933 0
1.7 0 2.38920 0 3.17647 0 −28.0947 0 −21.2917 0
1.8 0 7.02483 0 −2.19045 0 5.10018 0 22.3483 0
1.9 0 7.63776 0 7.65015 0 5.64550 0 31.3353 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
\(2\) \(1\)
\(29\) \(-1\)

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 1856.4.a.bf 9
4.b odd 2 1 1856.4.a.bg 9
8.b even 2 1 928.4.a.e yes 9
8.d odd 2 1 928.4.a.d ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
928.4.a.d ✓ 9 8.d odd 2 1
928.4.a.e yes 9 8.b even 2 1
1856.4.a.bf 9 1.a even 1 1 trivial
1856.4.a.bg 9 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1856))\):

\( T_{3}^{9} + 4 T_{3}^{8} - 138 T_{3}^{7} - 394 T_{3}^{6} + 5872 T_{3}^{5} + 10822 T_{3}^{4} + \cdots - 396802 \) Copy content Toggle raw display
\( T_{5}^{9} + 10 T_{5}^{8} - 568 T_{5}^{7} - 4532 T_{5}^{6} + 85426 T_{5}^{5} + 256400 T_{5}^{4} + \cdots - 37835002 \) Copy content Toggle raw display
\( T_{7}^{9} - 12 T_{7}^{8} - 1296 T_{7}^{7} + 17936 T_{7}^{6} + 407984 T_{7}^{5} - 6862272 T_{7}^{4} + \cdots + 1638662144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} + 4 T^{8} + \cdots - 396802 \) Copy content Toggle raw display
$5$ \( T^{9} + 10 T^{8} + \cdots - 37835002 \) Copy content Toggle raw display
$7$ \( T^{9} + \cdots + 1638662144 \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots + 86950571182 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots + 119297288098 \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots + 239296714517504 \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots + 71\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{9} + \cdots - 60\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( (T - 29)^{9} \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots + 12\!\cdots\!34 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots - 25\!\cdots\!12 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots + 71\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots + 30\!\cdots\!42 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots + 20\!\cdots\!22 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots + 25\!\cdots\!86 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots - 70\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots - 19\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots + 33\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots + 55\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots - 61\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots - 16\!\cdots\!86 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots - 77\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots - 37\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots + 21\!\cdots\!88 \) Copy content Toggle raw display
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