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Properties

Label 9248.2.a.bz
Level $9248$
Weight $2$
Character orbit 9248.a
Self dual yes
Analytic conductor $73.846$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9248,2,Mod(1,9248)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9248, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9248.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9248 = 2^{5} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9248.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: \(73.8456517893\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 40 x^{18} + 620 x^{16} - 4784 x^{14} + 19585 x^{12} - 41912 x^{10} + 43536 x^{8} - 20328 x^{6} + \cdots + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 544)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{17} q^{3} + \beta_{5} q^{5} - \beta_{12} q^{7} + (\beta_{11} - \beta_1 + 1) q^{9} + (\beta_{15} + \beta_{8}) q^{11} + \beta_{4} q^{13} + (\beta_{10} + \beta_{6} - \beta_{2} + 1) q^{15} + (\beta_{9} + \beta_{4} - \beta_1 + 2) q^{19}+ \cdots + ( - \beta_{19} + \beta_{18} + \cdots + 5 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 28 q^{9} - 8 q^{13} + 16 q^{15} + 40 q^{19} - 32 q^{21} + 28 q^{25} + 32 q^{35} + 40 q^{43} + 32 q^{47} + 36 q^{49} - 40 q^{53} + 48 q^{55} + 8 q^{59} + 72 q^{67} - 48 q^{69} - 48 q^{77} + 36 q^{81}+ \cdots - 32 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 40 x^{18} + 620 x^{16} - 4784 x^{14} + 19585 x^{12} - 41912 x^{10} + 43536 x^{8} - 20328 x^{6} + \cdots + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 18267628383 \nu^{18} - 712612199635 \nu^{16} + 10611913003840 \nu^{14} + \cdots + 4221265576528 ) / 3351525454031 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 43668805 \nu^{18} - 1730003806 \nu^{16} + 26409256133 \nu^{14} - 198712222860 \nu^{12} + \cdots - 4515483228 ) / 2587051682 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 425592862117 \nu^{19} - 17122608247099 \nu^{17} + 267772729570432 \nu^{15} + \cdots - 120989130998088 \nu ) / 13406101816124 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 544973673573 \nu^{18} + 21832474426316 \nu^{16} - 339189023530384 \nu^{14} + \cdots + 51676300354792 ) / 13406101816124 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 729687715819 \nu^{19} - 29333053236074 \nu^{17} + 458219040442108 \nu^{15} + \cdots - 414203996686196 \nu ) / 13406101816124 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1436420999437 \nu^{18} - 57094323940630 \nu^{16} + 876194755443248 \nu^{14} + \cdots - 99765636786336 ) / 13406101816124 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1620855033717 \nu^{18} - 64412053185194 \nu^{16} + 988179370693514 \nu^{14} + \cdots - 42588535718840 ) / 13406101816124 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1685039926147 \nu^{19} + 67375773796993 \nu^{17} + \cdots + 304851294440952 \nu ) / 13406101816124 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2046788438805 \nu^{18} + 81582745628052 \nu^{16} + \cdots + 213572676224496 ) / 13406101816124 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2871602179185 \nu^{18} + 114236722549248 \nu^{16} + \cdots + 43755049529316 ) / 13406101816124 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2923248676959 \nu^{18} + 116346135067952 \nu^{16} + \cdots + 84121790163792 ) / 13406101816124 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3144328235799 \nu^{19} - 125283900412380 \nu^{17} + \cdots + 48323560456800 \nu ) / 13406101816124 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3168289129224 \nu^{19} + 126499076996855 \nu^{17} + \cdots + 135397870819304 \nu ) / 13406101816124 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1863891780149 \nu^{18} + 74341924992857 \nu^{16} + \cdots + 150860303678668 ) / 6703050908062 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1937007975809 \nu^{19} - 77308476824227 \nu^{17} + \cdots - 169534116298574 \nu ) / 6703050908062 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 4301665475689 \nu^{19} - 172076624343895 \nu^{17} + \cdots - 887727469233332 \nu ) / 13406101816124 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 4462056211786 \nu^{19} + 177788805202205 \nu^{17} + \cdots + 324813255732684 \nu ) / 13406101816124 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 4736615570669 \nu^{19} - 188183737861468 \nu^{17} + \cdots + 463258373119348 \nu ) / 13406101816124 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 7131072695389 \nu^{19} - 283913057209206 \nu^{17} + \cdots - 474968396446528 \nu ) / 13406101816124 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - \beta_{12} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{14} - 2\beta_{9} - \beta_{6} - \beta_{4} + 2\beta_{2} + \beta _1 + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{18} + 3 \beta_{17} + 4 \beta_{16} + 7 \beta_{15} + \beta_{13} - 12 \beta_{12} + \cdots - 3 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11 \beta_{14} - 7 \beta_{11} + 7 \beta_{10} - 20 \beta_{9} + 2 \beta_{7} - 13 \beta_{6} - 15 \beta_{4} + \cdots + 75 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - \beta_{19} + 53 \beta_{18} + 63 \beta_{17} + 72 \beta_{16} + 70 \beta_{15} + 16 \beta_{13} + \cdots - 82 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 129 \beta_{14} - 133 \beta_{11} + 133 \beta_{10} - 225 \beta_{9} + 23 \beta_{7} - 166 \beta_{6} + \cdots + 863 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 7 \beta_{19} + 808 \beta_{18} + 1062 \beta_{17} + 1135 \beta_{16} + 829 \beta_{15} + \cdots - 1438 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1615 \beta_{14} - 2013 \beta_{11} + 2009 \beta_{10} - 2752 \beta_{9} + 216 \beta_{7} - 2191 \beta_{6} + \cdots + 10829 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 16 \beta_{19} + 11858 \beta_{18} + 16454 \beta_{17} + 16987 \beta_{16} + 10680 \beta_{15} + \cdots - 22155 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 21019 \beta_{14} - 28641 \beta_{11} + 28587 \beta_{10} - 35349 \beta_{9} + 1985 \beta_{7} + \cdots + 142005 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1202 \beta_{19} + 171049 \beta_{18} + 244075 \beta_{17} + 247384 \beta_{16} + 143293 \beta_{15} + \cdots - 324571 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 280144 \beta_{14} - 399842 \beta_{11} + 399669 \beta_{10} - 467813 \beta_{9} + 19103 \beta_{7} + \cdots + 1908467 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 23339 \beta_{19} + 2441745 \beta_{18} + 3533771 \beta_{17} + 3546328 \beta_{16} + \cdots - 4648674 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 3792099 \beta_{14} - 5553463 \beta_{11} + 5561691 \beta_{10} - 6309647 \beta_{9} + 198981 \beta_{7} + \cdots + 26023561 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 360477 \beta_{19} + 34614416 \beta_{18} + 50450762 \beta_{17} + 50356841 \beta_{16} + \cdots - 65844712 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 51867823 \beta_{14} - 77100301 \beta_{11} + 77356391 \beta_{10} - 86153110 \beta_{9} + 2258370 \beta_{7} + \cdots + 358019543 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 5135992 \beta_{19} + 488372686 \beta_{18} + 714324434 \beta_{17} + 710861593 \beta_{16} + \cdots - 927213153 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 714454449 \beta_{14} - 1071495967 \beta_{11} + 1076700541 \beta_{10} - 1185794883 \beta_{9} + \cdots + 4952897155 ) / 2 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 71011566 \beta_{19} + 6868163103 \beta_{18} + 10063509365 \beta_{17} + 9997832388 \beta_{16} + \cdots - 13014947921 \beta_{3} ) / 2 \) 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
−0.477107
2.62496
3.02416
−0.646193
−2.12443
0.373007
1.13075
−0.187649
1.83886
−3.73782
3.73782
−1.83886
0.187649
−1.13075
−0.373007
2.12443
0.646193
−3.02416
−2.62496
0.477107
0 −3.39376 0 −2.27168 0 −0.186067 0 8.51759 0
1.2 0 −3.02611 0 2.64460 0 4.15934 0 6.15732 0
1.3 0 −2.64768 0 −4.08952 0 3.94179 0 4.01023 0
1.4 0 −2.23930 0 −3.29169 0 −2.75910 0 2.01445 0
1.5 0 −2.15620 0 1.57755 0 2.46548 0 1.64918 0
1.6 0 −1.49030 0 −0.0537302 0 −4.62828 0 −0.779003 0
1.7 0 −1.32686 0 0.220143 0 −2.72498 0 −1.23945 0
1.8 0 −1.16495 0 3.58322 0 2.45360 0 −1.64289 0
1.9 0 −1.14381 0 1.71166 0 1.95318 0 −1.69170 0
1.10 0 −0.0654102 0 2.44478 0 −1.65973 0 −2.99572 0
1.11 0 0.0654102 0 −2.44478 0 1.65973 0 −2.99572 0
1.12 0 1.14381 0 −1.71166 0 −1.95318 0 −1.69170 0
1.13 0 1.16495 0 −3.58322 0 −2.45360 0 −1.64289 0
1.14 0 1.32686 0 −0.220143 0 2.72498 0 −1.23945 0
1.15 0 1.49030 0 0.0537302 0 4.62828 0 −0.779003 0
1.16 0 2.15620 0 −1.57755 0 −2.46548 0 1.64918 0
1.17 0 2.23930 0 3.29169 0 2.75910 0 2.01445 0
1.18 0 2.64768 0 4.08952 0 −3.94179 0 4.01023 0
1.19 0 3.02611 0 −2.64460 0 −4.15934 0 6.15732 0
1.20 0 3.39376 0 2.27168 0 0.186067 0 8.51759 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(17\) \( -1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9248.2.a.bz 20
4.b odd 2 1 9248.2.a.by 20
17.b even 2 1 inner 9248.2.a.bz 20
17.e odd 16 2 544.2.bb.f yes 20
68.d odd 2 1 9248.2.a.by 20
68.i even 16 2 544.2.bb.e ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
544.2.bb.e ✓ 20 68.i even 16 2
544.2.bb.f yes 20 17.e odd 16 2
9248.2.a.by 20 4.b odd 2 1
9248.2.a.by 20 68.d odd 2 1
9248.2.a.bz 20 1.a even 1 1 trivial
9248.2.a.bz 20 17.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9248))\):

\( T_{3}^{20} - 44 T_{3}^{18} + 806 T_{3}^{16} - 8032 T_{3}^{14} + 47788 T_{3}^{12} - 175472 T_{3}^{10} + \cdots + 512 \) Copy content Toggle raw display
\( T_{5}^{20} - 64 T_{5}^{18} + 1704 T_{5}^{16} - 24608 T_{5}^{14} + 210520 T_{5}^{12} - 1091520 T_{5}^{10} + \cdots + 512 \) Copy content Toggle raw display
\( T_{7}^{20} - 88 T_{7}^{18} + 3268 T_{7}^{16} - 67136 T_{7}^{14} + 840932 T_{7}^{12} - 6669600 T_{7}^{10} + \cdots + 4333568 \) Copy content Toggle raw display
\( T_{19}^{10} - 20 T_{19}^{9} + 80 T_{19}^{8} + 808 T_{19}^{7} - 7836 T_{19}^{6} + 18592 T_{19}^{5} + \cdots - 61696 \) Copy content Toggle raw display
\( T_{43}^{10} - 20 T_{43}^{9} + 6 T_{43}^{8} + 2648 T_{43}^{7} - 24516 T_{43}^{6} + 73136 T_{43}^{5} + \cdots - 1146752 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} - 44 T^{18} + \cdots + 512 \) Copy content Toggle raw display
$5$ \( T^{20} - 64 T^{18} + \cdots + 512 \) Copy content Toggle raw display
$7$ \( T^{20} - 88 T^{18} + \cdots + 4333568 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 890757632 \) Copy content Toggle raw display
$13$ \( (T^{10} + 4 T^{9} + \cdots - 256)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} \) Copy content Toggle raw display
$19$ \( (T^{10} - 20 T^{9} + \cdots - 61696)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 350599282688 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 5859680768 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 361943932928 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 490319316992 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 51\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( (T^{10} - 20 T^{9} + \cdots - 1146752)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} - 16 T^{9} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + 20 T^{9} + \cdots - 4771904)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} - 4 T^{9} + \cdots - 39810944)^{2} \) Copy content Toggle raw display
$61$ \( T^{20} - 368 T^{18} + \cdots + 19668992 \) Copy content Toggle raw display
$67$ \( (T^{10} - 36 T^{9} + \cdots - 647168)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 3969254555648 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 4552106528 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 51\!\cdots\!32 \) Copy content Toggle raw display
$83$ \( (T^{10} - 12 T^{9} + \cdots - 664544128)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} - 32 T^{9} + \cdots + 105060224)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 77\!\cdots\!72 \) Copy content Toggle raw display
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