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Properties

Label 1161.4.a.b
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: \( x^{18} - 93 x^{16} + 3500 x^{14} - 68750 x^{12} + 758449 x^{10} - 4705529 x^{8} + 15660730 x^{6} + \cdots - 4513968 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{6} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} - \beta_{13} q^{5} + (\beta_{8} - 3) q^{7} + (\beta_{13} + \beta_{12} + 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} - \beta_{13} q^{5} + (\beta_{8} - 3) q^{7} + (\beta_{13} + \beta_{12} + 2 \beta_1) q^{8} + ( - \beta_{8} - \beta_{5} - \beta_{2}) q^{10} + (\beta_{3} - 2 \beta_1) q^{11} + ( - \beta_{4} - \beta_{2}) q^{13} + (\beta_{13} - \beta_{12} + \cdots - 5 \beta_1) q^{14}+ \cdots + ( - 3 \beta_{17} + 11 \beta_{16} + \cdots - 182 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 42 q^{4} - 48 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 42 q^{4} - 48 q^{7} - 10 q^{10} + 2 q^{13} - 14 q^{16} - 244 q^{19} - 312 q^{22} - 60 q^{25} - 392 q^{28} - 530 q^{31} - 388 q^{34} - 136 q^{37} - 1162 q^{40} - 774 q^{43} - 522 q^{46} - 1326 q^{49} - 1298 q^{52} - 672 q^{55} - 2908 q^{58} - 1698 q^{61} - 1498 q^{64} - 2926 q^{67} - 2838 q^{70} - 256 q^{73} - 3804 q^{76} - 3458 q^{79} - 1386 q^{82} - 3726 q^{85} - 4192 q^{88} - 1644 q^{91} - 5182 q^{94} - 2892 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 93 x^{16} + 3500 x^{14} - 68750 x^{12} + 758449 x^{10} - 4705529 x^{8} + 15660730 x^{6} + \cdots - 4513968 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 619900 \nu^{17} - 2354262 \nu^{15} + 2435722531 \nu^{13} - 89733464197 \nu^{11} + \cdots - 32241017136600 \nu ) / 1019248582716 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10835053 \nu^{16} + 688025844 \nu^{14} - 12529431608 \nu^{12} - 28670161858 \nu^{10} + \cdots + 109404399184848 ) / 603999160128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11140453 \nu^{16} + 1097581308 \nu^{14} - 43942071200 \nu^{12} + 919692434726 \nu^{10} + \cdots - 147909664091376 ) / 603999160128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1553710 \nu^{16} + 166707999 \nu^{14} - 7173468755 \nu^{12} + 157928315675 \nu^{10} + \cdots - 15060596118480 ) / 75499895016 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5464679 \nu^{16} + 425032444 \nu^{14} - 12251000568 \nu^{12} + 154243520938 \nu^{10} + \cdots + 58366350326448 ) / 201333053376 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9975265 \nu^{16} - 907556112 \nu^{14} + 33179631692 \nu^{12} - 626828900438 \nu^{10} + \cdots + 56547201581424 ) / 150999790032 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 45545639 \nu^{16} - 4216852452 \nu^{14} + 157801324048 \nu^{12} - 3074141232994 \nu^{10} + \cdots + 346159643354064 ) / 603999160128 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 19811343 \nu^{16} + 1856636804 \nu^{14} - 70391010736 \nu^{12} + 1390089023154 \nu^{10} + \cdots - 157668923229840 ) / 201333053376 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 15224375 \nu^{17} + 1371407676 \nu^{15} - 49679082664 \nu^{13} + \cdots - 89198971908048 \nu ) / 1164855523104 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 432093685 \nu^{17} + 39408176316 \nu^{15} - 1443946532720 \nu^{13} + \cdots - 25\!\cdots\!20 \nu ) / 16307977323456 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 432093685 \nu^{17} - 39408176316 \nu^{15} + 1443946532720 \nu^{13} + \cdots + 22\!\cdots\!12 \nu ) / 16307977323456 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 334620155 \nu^{17} + 31965602856 \nu^{15} - 1232460494656 \nu^{13} + \cdots - 20\!\cdots\!24 \nu ) / 8153988661728 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 709743649 \nu^{17} - 64767472932 \nu^{15} + 2378706351368 \nu^{13} + \cdots + 42\!\cdots\!44 \nu ) / 5435992441152 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 2926647823 \nu^{17} + 269545535076 \nu^{15} - 10008500564912 \nu^{13} + \cdots - 18\!\cdots\!20 \nu ) / 16307977323456 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 994291001 \nu^{17} + 92699773260 \nu^{15} - 3491875868752 \nu^{13} + \cdots - 80\!\cdots\!56 \nu ) / 5435992441152 \) 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
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{12} + 18\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{6} + 2\beta_{5} + \beta_{4} + 24\beta_{2} + 175 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{17} + \beta_{16} + 2\beta_{14} + 36\beta_{13} + 32\beta_{12} + 2\beta_{11} + 2\beta_{3} + 375\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{10} + 33\beta_{9} - 2\beta_{8} + 35\beta_{6} + 76\beta_{5} + 35\beta_{4} + 564\beta_{2} + 3597 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 39 \beta_{17} + 31 \beta_{16} - 12 \beta_{15} + 90 \beta_{14} + 1052 \beta_{13} + 876 \beta_{12} + \cdots + 8393 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 186 \beta_{10} + 837 \beta_{9} - 24 \beta_{8} + 52 \beta_{7} + 947 \beta_{6} + 2278 \beta_{5} + \cdots + 79899 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1181 \beta_{17} + 665 \beta_{16} - 824 \beta_{15} + 3042 \beta_{14} + 28936 \beta_{13} + \cdots + 195859 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 6276 \beta_{10} + 19297 \beta_{9} + 1206 \beta_{8} + 3336 \beta_{7} + 23343 \beta_{6} + \cdots + 1855961 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 32763 \beta_{17} + 10467 \beta_{16} - 36604 \beta_{15} + 92698 \beta_{14} + 776176 \beta_{13} + \cdots + 4685717 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 188734 \beta_{10} + 423465 \beta_{9} + 86840 \beta_{8} + 141984 \beta_{7} + 550223 \beta_{6} + \cdots + 44265127 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 874309 \beta_{17} + 66513 \beta_{16} - 1343024 \beta_{15} + 2685586 \beta_{14} + \cdots + 113798827 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 5378424 \beta_{10} + 8986077 \beta_{9} + 3656414 \beta_{8} + 5062388 \beta_{7} + 12657931 \beta_{6} + \cdots + 1072550941 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 22892043 \beta_{17} - 3612973 \beta_{16} - 44327804 \beta_{15} + 75544826 \beta_{14} + \cdots + 2789943837 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 148834050 \beta_{10} + 185081965 \beta_{9} + 127851544 \beta_{8} + 163790660 \beta_{7} + \cdots + 26244864203 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 593494429 \beta_{17} - 223322551 \beta_{16} - 1369935032 \beta_{15} + 2084104882 \beta_{14} + \cdots + 68831289395 \beta_1 \) 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
−5.05367
−4.75435
−3.67383
−3.55929
−3.31821
−1.94994
−1.60062
−0.860273
−0.759009
0.759009
0.860273
1.60062
1.94994
3.31821
3.55929
3.67383
4.75435
5.05367
−5.05367 0 17.5396 15.3057 0 −4.05150 −48.2100 0 −77.3499
1.2 −4.75435 0 14.6039 −8.09682 0 −18.2029 −31.3970 0 38.4951
1.3 −3.67383 0 5.49703 0.536107 0 0.259201 9.19549 0 −1.96957
1.4 −3.55929 0 4.66857 −7.68841 0 −2.28997 11.8575 0 27.3653
1.5 −3.31821 0 3.01051 −1.68336 0 22.3256 16.5562 0 5.58575
1.6 −1.94994 0 −4.19773 −18.3632 0 −24.2340 23.7849 0 35.8071
1.7 −1.60062 0 −5.43800 17.2244 0 0.505703 21.5092 0 −27.5697
1.8 −0.860273 0 −7.25993 −2.30688 0 −21.3318 13.1277 0 1.98455
1.9 −0.759009 0 −7.42390 9.68188 0 23.0197 11.7069 0 −7.34864
1.10 0.759009 0 −7.42390 −9.68188 0 23.0197 −11.7069 0 −7.34864
1.11 0.860273 0 −7.25993 2.30688 0 −21.3318 −13.1277 0 1.98455
1.12 1.60062 0 −5.43800 −17.2244 0 0.505703 −21.5092 0 −27.5697
1.13 1.94994 0 −4.19773 18.3632 0 −24.2340 −23.7849 0 35.8071
1.14 3.31821 0 3.01051 1.68336 0 22.3256 −16.5562 0 5.58575
1.15 3.55929 0 4.66857 7.68841 0 −2.28997 −11.8575 0 27.3653
1.16 3.67383 0 5.49703 −0.536107 0 0.259201 −9.19549 0 −1.96957
1.17 4.75435 0 14.6039 8.09682 0 −18.2029 31.3970 0 38.4951
1.18 5.05367 0 17.5396 −15.3057 0 −4.05150 48.2100 0 −77.3499
\(n\): 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.b ✓ 18
3.b odd 2 1 inner 1161.4.a.b ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1161.4.a.b ✓ 18 1.a even 1 1 trivial
1161.4.a.b ✓ 18 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} - 93 T_{2}^{16} + 3500 T_{2}^{14} - 68750 T_{2}^{12} + 758449 T_{2}^{10} - 4705529 T_{2}^{8} + \cdots - 4513968 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1161))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 93 T^{16} + \cdots - 4513968 \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots - 36899186535472 \) Copy content Toggle raw display
$7$ \( (T^{9} + 24 T^{8} + \cdots + 5881329)^{2} \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots - 22\!\cdots\!68 \) Copy content Toggle raw display
$13$ \( (T^{9} + \cdots + 1515785710311)^{2} \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots - 56\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( (T^{9} + \cdots - 15\!\cdots\!72)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 22\!\cdots\!28 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots - 57\!\cdots\!28 \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots + 44\!\cdots\!32)^{2} \) Copy content Toggle raw display
$37$ \( (T^{9} + \cdots + 41\!\cdots\!96)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 47\!\cdots\!48 \) Copy content Toggle raw display
$43$ \( (T + 43)^{18} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 22\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots - 37\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 10\!\cdots\!12 \) Copy content Toggle raw display
$61$ \( (T^{9} + \cdots - 44\!\cdots\!64)^{2} \) Copy content Toggle raw display
$67$ \( (T^{9} + \cdots + 41\!\cdots\!12)^{2} \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 99\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( (T^{9} + \cdots + 17\!\cdots\!28)^{2} \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots + 30\!\cdots\!56)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 17\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 10\!\cdots\!92 \) Copy content Toggle raw display
$97$ \( (T^{9} + \cdots - 54\!\cdots\!27)^{2} \) Copy content Toggle raw display
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