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Properties

Label 111.3.d.c
Level $111$
Weight $3$
Character orbit 111.d
Analytic conductor $3.025$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [111,3,Mod(110,111)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(111, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("111.110");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 111 = 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 111.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(3.02453093440\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 8 x^{14} + 536 x^{12} - 302 x^{11} + 6787 x^{10} - 6032 x^{9} + 110073 x^{8} - 140264 x^{7} + \cdots + 228357516 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} + \beta_{5} q^{3} + (\beta_{6} + 1) q^{4} - \beta_{2} q^{5} - \beta_{3} q^{6} + ( - \beta_{9} + \beta_{6} - 1) q^{7} + \beta_{15} q^{8} + (\beta_{14} - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{2} + \beta_{5} q^{3} + (\beta_{6} + 1) q^{4} - \beta_{2} q^{5} - \beta_{3} q^{6} + ( - \beta_{9} + \beta_{6} - 1) q^{7} + \beta_{15} q^{8} + (\beta_{14} - 4) q^{9} + (\beta_{9} - \beta_{7} + \beta_{5}) q^{10} - \beta_{11} q^{11} + ( - \beta_{12} + \beta_{9} + \beta_{7}) q^{12} + ( - \beta_{13} + \beta_{3}) q^{13} + (2 \beta_{15} + \beta_{8} + \cdots + \beta_1) q^{14}+ \cdots + (4 \beta_{11} + 13 \beta_{9} + \cdots + 26) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{4} - 16 q^{7} - 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 12 q^{4} - 16 q^{7} - 64 q^{9} - 4 q^{10} - 2 q^{12} - 36 q^{16} - 26 q^{21} - 60 q^{25} - 54 q^{27} + 172 q^{28} + 104 q^{30} + 6 q^{33} + 220 q^{34} - 138 q^{36} + 132 q^{37} + 84 q^{40} + 200 q^{46} - 46 q^{48} - 248 q^{49} - 188 q^{58} - 170 q^{63} - 484 q^{64} + 428 q^{67} - 384 q^{70} - 76 q^{73} + 358 q^{75} + 294 q^{78} - 496 q^{81} - 400 q^{84} + 272 q^{85} + 106 q^{90} + 312 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 8 x^{14} + 536 x^{12} - 302 x^{11} + 6787 x^{10} - 6032 x^{9} + 110073 x^{8} - 140264 x^{7} + \cdots + 228357516 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 15\!\cdots\!71 \nu^{15} + \cdots + 10\!\cdots\!00 ) / 11\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 77\!\cdots\!31 \nu^{15} + \cdots - 92\!\cdots\!40 ) / 11\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 57\!\cdots\!50 \nu^{15} + \cdots - 42\!\cdots\!20 ) / 34\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13\!\cdots\!11 \nu^{15} + \cdots - 74\!\cdots\!32 ) / 68\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14\!\cdots\!63 \nu^{15} + \cdots + 10\!\cdots\!96 ) / 68\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12\!\cdots\!42 \nu^{15} + \cdots - 14\!\cdots\!00 ) / 30\!\cdots\!21 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 32\!\cdots\!17 \nu^{15} + \cdots + 58\!\cdots\!56 ) / 68\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\!\cdots\!23 \nu^{15} + \cdots - 33\!\cdots\!68 ) / 22\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 17\!\cdots\!12 \nu^{15} + \cdots - 62\!\cdots\!38 ) / 30\!\cdots\!21 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 46\!\cdots\!71 \nu^{15} + \cdots - 12\!\cdots\!72 ) / 68\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 18\!\cdots\!71 \nu^{15} + \cdots + 27\!\cdots\!64 ) / 22\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 54\!\cdots\!95 \nu^{15} + \cdots - 50\!\cdots\!06 ) / 57\!\cdots\!79 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 18\!\cdots\!48 \nu^{15} + \cdots - 60\!\cdots\!20 ) / 17\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 19\!\cdots\!42 \nu^{15} + \cdots + 27\!\cdots\!95 ) / 17\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 43\!\cdots\!73 \nu^{15} + \cdots + 20\!\cdots\!36 ) / 22\!\cdots\!16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} - \beta_{8} - 6\beta_{5} - 2\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{15} + 3 \beta_{14} + \beta_{13} + \beta_{10} - \beta_{9} - 3 \beta_{8} - \beta_{7} + \beta_{6} + \cdots - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 20 \beta_{15} + 3 \beta_{13} - 15 \beta_{10} + 26 \beta_{8} - 54 \beta_{7} - 6 \beta_{5} + \cdots + 3 \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 14 \beta_{15} + 18 \beta_{14} + 4 \beta_{13} + 12 \beta_{12} + 3 \beta_{11} + 4 \beta_{10} + 12 \beta_{9} + \cdots - 378 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 235 \beta_{15} - 38 \beta_{14} - 15 \beta_{13} - 58 \beta_{12} - 58 \beta_{11} + 75 \beta_{10} + \cdots + 450 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 272 \beta_{15} - 573 \beta_{14} - 209 \beta_{13} - 405 \beta_{12} - 54 \beta_{11} - 209 \beta_{10} + \cdots - 2979 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3737 \beta_{15} + 2522 \beta_{14} - 1134 \beta_{13} - 476 \beta_{12} - 860 \beta_{11} + 4788 \beta_{10} + \cdots + 10458 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1580 \beta_{15} - 2479 \beta_{14} - 764 \beta_{13} - 1314 \beta_{12} - 458 \beta_{11} - 1620 \beta_{10} + \cdots + 27137 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 85648 \beta_{15} + 22098 \beta_{14} + 11325 \beta_{13} + 16080 \beta_{12} + 14766 \beta_{11} + \cdots - 128898 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 57703 \beta_{15} + 93564 \beta_{14} + 47938 \beta_{13} + 129441 \beta_{12} + 17472 \beta_{11} + \cdots + 1485894 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 728315 \beta_{15} - 760590 \beta_{14} + 337755 \beta_{13} + 364554 \beta_{12} + 625746 \beta_{11} + \cdots - 6645090 ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 1279760 \beta_{15} + 2738664 \beta_{14} + 876262 \beta_{13} + 1834815 \beta_{12} + 168051 \beta_{11} + \cdots - 16423902 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 31374121 \beta_{15} - 13101320 \beta_{14} - 3283410 \beta_{13} - 5828386 \beta_{12} - 2303548 \beta_{11} + \cdots + 3664668 ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 17295617 \beta_{15} - 8196129 \beta_{14} - 11576489 \beta_{13} - 33411993 \beta_{12} - 3586989 \beta_{11} + \cdots - 534549549 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 129059846 \beta_{15} + 144978110 \beta_{14} - 125815371 \beta_{13} - 168775790 \beta_{12} + \cdots + 2564380530 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/111\mathbb{Z}\right)^\times\).

\(n\) \(38\) \(76\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
110.1
3.16398 + 2.87898i
3.16398 − 2.87898i
−3.25387 + 2.44177i
−3.25387 − 2.44177i
1.81922 + 2.08250i
1.81922 − 2.08250i
0.793434 + 2.72258i
0.793434 − 2.72258i
−3.31342 + 2.72258i
−3.31342 − 2.72258i
2.49967 + 2.08250i
2.49967 − 2.08250i
−0.231978 + 2.44177i
−0.231978 − 2.44177i
−1.47703 + 2.87898i
−1.47703 − 2.87898i
−3.21328 −0.843475 − 2.87898i 6.32514 1.61921 2.71032 + 9.25097i 7.84115 −7.47133 −7.57710 + 4.85670i −5.20296
110.2 −3.21328 −0.843475 + 2.87898i 6.32514 1.61921 2.71032 − 9.25097i 7.84115 −7.47133 −7.57710 − 4.85670i −5.20296
110.3 −2.59436 1.74292 − 2.44177i 2.73070 −4.18351 −4.52177 + 6.33482i −6.63698 3.29301 −2.92443 − 8.51162i 10.8535
110.4 −2.59436 1.74292 + 2.44177i 2.73070 −4.18351 −4.52177 − 6.33482i −6.63698 3.29301 −2.92443 + 8.51162i 10.8535
110.5 −1.32152 −2.15944 − 2.08250i −2.25359 2.45068 2.85374 + 2.75206i −5.33386 8.26423 0.326371 + 8.99408i −3.23862
110.6 −1.32152 −2.15944 + 2.08250i −2.25359 2.45068 2.85374 − 2.75206i −5.33386 8.26423 0.326371 − 8.99408i −3.23862
110.7 −0.444687 1.25999 − 2.72258i −3.80225 7.67272 −0.560303 + 1.21069i 0.129691 3.46956 −5.82484 − 6.86085i −3.41196
110.8 −0.444687 1.25999 + 2.72258i −3.80225 7.67272 −0.560303 − 1.21069i 0.129691 3.46956 −5.82484 + 6.86085i −3.41196
110.9 0.444687 1.25999 − 2.72258i −3.80225 −7.67272 0.560303 − 1.21069i 0.129691 −3.46956 −5.82484 − 6.86085i −3.41196
110.10 0.444687 1.25999 + 2.72258i −3.80225 −7.67272 0.560303 + 1.21069i 0.129691 −3.46956 −5.82484 + 6.86085i −3.41196
110.11 1.32152 −2.15944 − 2.08250i −2.25359 −2.45068 −2.85374 − 2.75206i −5.33386 −8.26423 0.326371 + 8.99408i −3.23862
110.12 1.32152 −2.15944 + 2.08250i −2.25359 −2.45068 −2.85374 + 2.75206i −5.33386 −8.26423 0.326371 − 8.99408i −3.23862
110.13 2.59436 1.74292 − 2.44177i 2.73070 4.18351 4.52177 − 6.33482i −6.63698 −3.29301 −2.92443 − 8.51162i 10.8535
110.14 2.59436 1.74292 + 2.44177i 2.73070 4.18351 4.52177 + 6.33482i −6.63698 −3.29301 −2.92443 + 8.51162i 10.8535
110.15 3.21328 −0.843475 − 2.87898i 6.32514 −1.61921 −2.71032 − 9.25097i 7.84115 7.47133 −7.57710 + 4.85670i −5.20296
110.16 3.21328 −0.843475 + 2.87898i 6.32514 −1.61921 −2.71032 + 9.25097i 7.84115 7.47133 −7.57710 − 4.85670i −5.20296
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 110.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 111.3.d.c ✓ 16
3.b odd 2 1 inner 111.3.d.c ✓ 16
37.b even 2 1 inner 111.3.d.c ✓ 16
111.d odd 2 1 inner 111.3.d.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.3.d.c ✓ 16 1.a even 1 1 trivial
111.3.d.c ✓ 16 3.b odd 2 1 inner
111.3.d.c ✓ 16 37.b even 2 1 inner
111.3.d.c ✓ 16 111.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 19T_{2}^{6} + 103T_{2}^{4} - 141T_{2}^{2} + 24 \) acting on \(S_{3}^{\mathrm{new}}(111, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 19 T^{6} + \cdots + 24)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} + 16 T^{6} + \cdots + 6561)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} - 85 T^{6} + \cdots + 16224)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 4 T^{3} - 59 T^{2} + \cdots + 36)^{4} \) Copy content Toggle raw display
$11$ \( (T^{8} + 669 T^{6} + \cdots + 74118159)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 912 T^{6} + \cdots + 197648424)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 1237 T^{6} + \cdots + 414536064)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 2673 T^{6} + \cdots + 50597996544)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 2230 T^{6} + \cdots + 62365056)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 1297 T^{6} + \cdots + 631113216)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 2751 T^{6} + \cdots + 3162374784)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 3512479453921)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 11505806530524)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 21263808047616)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 32678399830464)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 4116 T^{6} + \cdots + 303587979264)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 17356 T^{6} + \cdots + 959078621184)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 14283 T^{6} + \cdots + 256152357504)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 107 T^{3} + \cdots - 1713672)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 32678399830464)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 19 T^{3} + \cdots + 277227)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 15\!\cdots\!24)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 31895712071424)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 68\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 12953087115264)^{2} \) Copy content Toggle raw display
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