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Properties

Label 1161.4.a.a
Level $1161$
Weight $4$
Character orbit 1161.a
Self dual yes
Analytic conductor $68.501$
Analytic rank $1$
Dimension $18$
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] = [1161,4,Mod(1,1161)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1161, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1161.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1161 = 3^{3} \cdot 43 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1161.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: \(68.5012175167\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 93 x^{16} + 3484 x^{14} - 67438 x^{12} + 720201 x^{10} - 4197505 x^{8} + 12282514 x^{6} + \cdots - 798768 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{8} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} - \beta_{11} q^{5} + (\beta_{8} - 4) q^{7} + (\beta_{3} + 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} - \beta_{11} q^{5} + (\beta_{8} - 4) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + ( - \beta_{6} + 2) q^{10} + (\beta_{15} + \beta_{11} - \beta_1) q^{11} + ( - \beta_{9} - \beta_{8} - \beta_{7} + \cdots - 6) q^{13}+ \cdots + (6 \beta_{17} - 2 \beta_{16} + \cdots - 9 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 42 q^{4} - 76 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 42 q^{4} - 76 q^{7} + 38 q^{10} - 118 q^{13} + 50 q^{16} - 84 q^{19} - 140 q^{22} - 12 q^{25} - 468 q^{28} - 418 q^{31} - 588 q^{34} - 684 q^{37} - 830 q^{40} + 774 q^{43} - 2478 q^{46} + 914 q^{49} - 2746 q^{52} - 2644 q^{55} - 2736 q^{58} - 2466 q^{61} - 3002 q^{64} - 146 q^{67} - 526 q^{70} - 1512 q^{73} - 1100 q^{76} - 3862 q^{79} - 4162 q^{82} - 2410 q^{85} - 1556 q^{88} - 5892 q^{91} + 946 q^{94} - 3060 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 93 x^{16} + 3484 x^{14} - 67438 x^{12} + 720201 x^{10} - 4197505 x^{8} + 12282514 x^{6} + \cdots - 798768 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 18\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 29827 \nu^{16} + 2659234 \nu^{14} - 94463186 \nu^{12} + 1709045844 \nu^{10} + \cdots - 138752398224 ) / 1403698896 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 69907 \nu^{16} + 6061108 \nu^{14} - 205164224 \nu^{12} + 3407847978 \nu^{10} + \cdots + 45028784304 ) / 1871598528 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 713291 \nu^{16} + 63812612 \nu^{14} - 2268650896 \nu^{12} + 40836612666 \nu^{10} + \cdots - 622368010512 ) / 16844386752 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 17225 \nu^{16} + 1634688 \nu^{14} - 62734844 \nu^{12} + 1248469498 \nu^{10} + \cdots - 55727225712 ) / 311933088 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 788759 \nu^{16} - 72648284 \nu^{14} + 2686261216 \nu^{12} - 51059013258 \nu^{10} + \cdots + 1720033709616 ) / 8422193376 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 318115 \nu^{16} - 29496836 \nu^{14} + 1099477408 \nu^{12} - 21095937994 \nu^{10} + \cdots + 696193395984 ) / 1871598528 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2012387 \nu^{16} - 186719204 \nu^{14} + 6969314032 \nu^{12} - 134090806026 \nu^{10} + \cdots + 5457664576080 ) / 5614795584 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 35317441 \nu^{17} + 3253850500 \nu^{15} - 120302022128 \nu^{13} + \cdots - 53848518630192 \nu ) / 724308630336 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 16347655 \nu^{17} - 1530402988 \nu^{15} + 57769672208 \nu^{13} - 1126934027970 \nu^{11} + \cdots + 27932112862416 \nu ) / 241436210112 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 49451047 \nu^{17} - 4550833252 \nu^{15} + 167985763352 \nu^{13} - 3182330485482 \nu^{11} + \cdots + 99204713576784 \nu ) / 724308630336 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 24295739 \nu^{17} - 2257352996 \nu^{15} + 84416278360 \nu^{13} - 1628336958162 \nu^{11} + \cdots + 80010889550928 \nu ) / 241436210112 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 126586885 \nu^{17} + 11722828660 \nu^{15} - 436583558432 \nu^{13} + \cdots - 330286126017456 \nu ) / 724308630336 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 3232808 \nu^{17} + 299738512 \nu^{15} - 11177944367 \nu^{13} + 214814280383 \nu^{11} + \cdots - 8439169339296 \nu ) / 10059842088 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 88628237 \nu^{17} + 8205690668 \nu^{15} - 305435838496 \nu^{13} + \cdots - 226510338424176 \nu ) / 241436210112 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 18\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} + 23\beta_{2} + 179 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{17} - 2\beta_{16} + \beta_{14} + 3\beta_{13} - \beta_{12} + 3\beta_{11} + 32\beta_{3} + 371\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{10} + 35\beta_{9} - 23\beta_{8} + 37\beta_{7} + 41\beta_{6} - 33\beta_{4} + 519\beta_{2} + 3675 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 74 \beta_{17} - 78 \beta_{16} - 6 \beta_{15} + 15 \beta_{14} + 129 \beta_{13} - 31 \beta_{12} + \cdots + 8107 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 100 \beta_{10} + 961 \beta_{9} - 385 \beta_{8} + 1035 \beta_{7} + 1273 \beta_{6} - 16 \beta_{5} + \cdots + 80083 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2038 \beta_{17} - 2198 \beta_{16} - 324 \beta_{15} + \beta_{14} + 4167 \beta_{13} - 713 \beta_{12} + \cdots + 182447 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3618 \beta_{10} + 24595 \beta_{9} - 4855 \beta_{8} + 26249 \beta_{7} + 35461 \beta_{6} + \cdots + 1798031 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 50762 \beta_{17} - 54742 \beta_{16} - 12246 \beta_{15} - 7333 \beta_{14} + 120261 \beta_{13} + \cdots + 4173863 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 114240 \beta_{10} + 612093 \beta_{9} - 20049 \beta_{8} + 637223 \beta_{7} + 933201 \beta_{6} + \cdots + 41045167 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1214142 \beta_{17} - 1283182 \beta_{16} - 403644 \beta_{15} - 302503 \beta_{14} + 3268671 \beta_{13} + \cdots + 96457647 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 3346118 \beta_{10} + 15022047 \beta_{9} + 1557489 \beta_{8} + 15136557 \beta_{7} + 23730069 \beta_{6} + \cdots + 946614043 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 28573282 \beta_{17} - 29044654 \beta_{16} - 12400662 \beta_{15} - 9209629 \beta_{14} + \cdots + 2244353007 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 93534532 \beta_{10} + 365675001 \beta_{9} + 80280279 \beta_{8} + 355415539 \beta_{7} + \cdots + 21982493379 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 668816054 \beta_{17} - 642080422 \beta_{16} - 364653588 \beta_{15} - 247118431 \beta_{14} + \cdots + 52476027567 \beta_1 \) 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
−4.90351
−4.81068
−4.27409
−3.40906
−2.70082
−2.56121
−1.19860
−0.597727
−0.524689
0.524689
0.597727
1.19860
2.56121
2.70082
3.40906
4.27409
4.81068
4.90351
−4.90351 0 16.0444 4.51675 0 8.38968 −39.4460 0 −22.1479
1.2 −4.81068 0 15.1426 6.42293 0 −18.7929 −34.3609 0 −30.8987
1.3 −4.27409 0 10.2678 −13.4423 0 −27.1657 −9.69295 0 57.4538
1.4 −3.40906 0 3.62169 0.511927 0 −8.68174 14.9259 0 −1.74519
1.5 −2.70082 0 −0.705576 −10.0551 0 30.6339 23.5122 0 27.1569
1.6 −2.56121 0 −1.44023 6.98549 0 13.0164 24.1784 0 −17.8913
1.7 −1.19860 0 −6.56336 −18.3531 0 −3.38481 17.4556 0 21.9980
1.8 −0.597727 0 −7.64272 18.1420 0 −34.3827 9.35007 0 −10.8439
1.9 −0.524689 0 −7.72470 7.77930 0 2.36782 8.25058 0 −4.08171
1.10 0.524689 0 −7.72470 −7.77930 0 2.36782 −8.25058 0 −4.08171
1.11 0.597727 0 −7.64272 −18.1420 0 −34.3827 −9.35007 0 −10.8439
1.12 1.19860 0 −6.56336 18.3531 0 −3.38481 −17.4556 0 21.9980
1.13 2.56121 0 −1.44023 −6.98549 0 13.0164 −24.1784 0 −17.8913
1.14 2.70082 0 −0.705576 10.0551 0 30.6339 −23.5122 0 27.1569
1.15 3.40906 0 3.62169 −0.511927 0 −8.68174 −14.9259 0 −1.74519
1.16 4.27409 0 10.2678 13.4423 0 −27.1657 9.69295 0 57.4538
1.17 4.81068 0 15.1426 −6.42293 0 −18.7929 34.3609 0 −30.8987
1.18 4.90351 0 16.0444 −4.51675 0 8.38968 39.4460 0 −22.1479
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(43\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1161.4.a.a ✓ 18
3.b odd 2 1 inner 1161.4.a.a ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1161.4.a.a ✓ 18 1.a even 1 1 trivial
1161.4.a.a ✓ 18 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} - 93 T_{2}^{16} + 3484 T_{2}^{14} - 67438 T_{2}^{12} + 720201 T_{2}^{10} - 4197505 T_{2}^{8} + \cdots - 798768 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1161))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 93 T^{16} + \cdots - 798768 \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots - 13\!\cdots\!52 \) Copy content Toggle raw display
$7$ \( (T^{9} + 38 T^{8} + \cdots + 4085851153)^{2} \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots - 87\!\cdots\!48 \) Copy content Toggle raw display
$13$ \( (T^{9} + \cdots + 1559223168089)^{2} \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots - 30\!\cdots\!28 \) Copy content Toggle raw display
$19$ \( (T^{9} + \cdots + 25\!\cdots\!04)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 11\!\cdots\!12 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots - 75\!\cdots\!68 \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots - 11\!\cdots\!48)^{2} \) Copy content Toggle raw display
$37$ \( (T^{9} + \cdots - 96\!\cdots\!88)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 15\!\cdots\!32 \) Copy content Toggle raw display
$43$ \( (T - 43)^{18} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 11\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots - 65\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 90\!\cdots\!88 \) Copy content Toggle raw display
$61$ \( (T^{9} + \cdots - 27\!\cdots\!56)^{2} \) Copy content Toggle raw display
$67$ \( (T^{9} + \cdots - 46\!\cdots\!04)^{2} \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 16\!\cdots\!08 \) Copy content Toggle raw display
$73$ \( (T^{9} + \cdots - 50\!\cdots\!28)^{2} \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots + 10\!\cdots\!72)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 22\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 38\!\cdots\!92 \) Copy content Toggle raw display
$97$ \( (T^{9} + \cdots + 19\!\cdots\!89)^{2} \) Copy content Toggle raw display
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