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Properties

Label 105.3.o.a
Level $105$
Weight $3$
Character orbit 105.o
Analytic conductor $2.861$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [105,3,Mod(44,105)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(105, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("105.44");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 105.o (of order \(6\), degree \(2\), 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: \(2.86104277578\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 4 x^{14} + 12 x^{13} + 162 x^{12} - 524 x^{11} - 88 x^{10} + 1492 x^{9} + \cdots + 1521 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + (\beta_{13} - \beta_{9} - \beta_{3}) q^{3} + (\beta_{10} - \beta_1 - 1) q^{4} + ( - \beta_{15} - \beta_{12} + \cdots - \beta_{2}) q^{5}+ \cdots + (2 \beta_{12} - 2 \beta_{11} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + (\beta_{13} - \beta_{9} - \beta_{3}) q^{3} + (\beta_{10} - \beta_1 - 1) q^{4} + ( - \beta_{15} - \beta_{12} + \cdots - \beta_{2}) q^{5}+ \cdots + ( - \beta_{14} - 10 \beta_{11} + \cdots + 30) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{4} - 80 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{4} - 80 q^{6} - 8 q^{9} - 40 q^{10} - 80 q^{15} + 32 q^{16} + 48 q^{19} - 8 q^{21} + 40 q^{30} + 344 q^{31} - 80 q^{34} + 496 q^{36} - 32 q^{39} + 120 q^{40} - 80 q^{45} - 120 q^{46} - 208 q^{49} - 40 q^{51} + 200 q^{54} + 40 q^{60} - 392 q^{61} - 544 q^{64} + 120 q^{66} - 240 q^{69} - 760 q^{70} + 200 q^{75} - 336 q^{76} + 608 q^{79} - 328 q^{81} - 344 q^{84} - 560 q^{85} + 80 q^{90} + 1088 q^{91} + 480 q^{94} - 400 q^{96} + 480 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 4 x^{14} + 12 x^{13} + 162 x^{12} - 524 x^{11} - 88 x^{10} + 1492 x^{9} + \cdots + 1521 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1282138 \nu^{15} + 19217450 \nu^{14} - 57698230 \nu^{13} - 201080763 \nu^{12} + \cdots + 29387553354 ) / 3843901530 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 945353839 \nu^{15} + 19734882097 \nu^{14} - 9615662912 \nu^{13} - 208584356535 \nu^{12} + \cdots - 29454867221481 ) / 2648448154170 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6799338422 \nu^{15} + 5815573694 \nu^{14} - 110020669305 \nu^{13} - 206978139649 \nu^{12} + \cdots + 36969480533544 ) / 2648448154170 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6939940550 \nu^{15} - 489734378 \nu^{14} - 79452033908 \nu^{13} - 161040415944 \nu^{12} + \cdots + 10360850614686 ) / 2648448154170 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 8517854635 \nu^{15} + 47901289016 \nu^{14} + 10808846529 \nu^{13} - 254360686162 \nu^{12} + \cdots + 26162339620443 ) / 2648448154170 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10954917349 \nu^{15} + 11841164258 \nu^{14} + 138130967169 \nu^{13} + \cdots - 31317616718193 ) / 2648448154170 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1683778105 \nu^{15} - 2653967414 \nu^{14} - 17809640997 \nu^{13} - 13603646729 \nu^{12} + \cdots - 1451250345714 ) / 378349736310 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 58793534 \nu^{15} - 247088261 \nu^{14} - 186701243 \nu^{13} + 925377984 \nu^{12} + \cdots - 169644585825 ) / 11081373030 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2053516175 \nu^{15} - 8267524636 \nu^{14} - 7910711853 \nu^{13} + 29044383254 \nu^{12} + \cdots - 5795645027601 ) / 378349736310 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 15731411194 \nu^{15} + 64536858778 \nu^{14} + 74366831386 \nu^{13} + \cdots + 33091548064380 ) / 2648448154170 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 394610 \nu^{15} + 1397252 \nu^{14} + 2812046 \nu^{13} - 4855143 \nu^{12} - 71847798 \nu^{11} + \cdots + 209664117 ) / 32307210 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 5555945704 \nu^{15} - 12683893446 \nu^{14} - 51058434102 \nu^{13} - 5680074782 \nu^{12} + \cdots - 550491674226 ) / 441408025695 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 34807975271 \nu^{15} + 87110547559 \nu^{14} + 303155003931 \nu^{13} + \cdots + 9989689919079 ) / 2648448154170 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 37024534763 \nu^{15} - 117279863141 \nu^{14} - 278878852346 \nu^{13} + 286708401207 \nu^{12} + \cdots - 41746013359011 ) / 2648448154170 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 62968587943 \nu^{15} + 235645881604 \nu^{14} + 369876667756 \nu^{13} + \cdots + 128382868343808 ) / 2648448154170 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{15} + \beta_{14} + \beta_{13} + 2 \beta_{11} + \beta_{10} + \beta_{9} + \beta_{7} + \beta_{5} + \cdots + 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{15} + \beta_{14} - 2 \beta_{13} - 3 \beta_{12} - \beta_{11} - 2 \beta_{10} + \beta_{9} + \cdots - 9 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 17 \beta_{15} - 7 \beta_{14} + 29 \beta_{13} + 18 \beta_{12} + 2 \beta_{11} + 13 \beta_{10} + \cdots + 51 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6 \beta_{15} + 12 \beta_{14} - 30 \beta_{13} - 30 \beta_{12} + 14 \beta_{11} - 5 \beta_{10} + 60 \beta_{9} + \cdots - 66 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 35 \beta_{15} + 5 \beta_{14} - 51 \beta_{13} - 60 \beta_{12} - 90 \beta_{11} - 45 \beta_{10} + \cdots + 225 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 188 \beta_{15} - 49 \beta_{14} + 620 \beta_{13} + 495 \beta_{12} + 74 \beta_{11} + 103 \beta_{10} + \cdots + 198 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 851 \beta_{15} + 185 \beta_{14} - 5865 \beta_{13} - 4872 \beta_{12} + 324 \beta_{11} - 237 \beta_{10} + \cdots - 1677 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 140 \beta_{15} + 1192 \beta_{14} + 7656 \beta_{13} + 6252 \beta_{12} - 770 \beta_{11} - 904 \beta_{10} + \cdots + 105 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 6493 \beta_{15} - 15521 \beta_{14} - 9587 \beta_{13} - 8064 \beta_{12} - 952 \beta_{11} + \cdots + 36681 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 7580 \beta_{15} + 24505 \beta_{14} - 55682 \beta_{13} - 46005 \beta_{12} + 19685 \beta_{11} + \cdots - 78336 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 56503 \beta_{15} - 30049 \beta_{14} + 570965 \beta_{13} + 468402 \beta_{12} - 189892 \beta_{11} + \cdots + 228525 ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 286602 \beta_{15} - 255672 \beta_{14} - 640926 \beta_{13} - 528462 \beta_{12} + 211166 \beta_{11} + \cdots + 681258 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 2083883 \beta_{15} + 3003875 \beta_{14} - 202827 \beta_{13} - 173628 \beta_{12} + 512796 \beta_{11} + \cdots - 10435617 ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1254626 \beta_{15} - 4365001 \beta_{14} + 5874734 \beta_{13} + 4823145 \beta_{12} - 4106908 \beta_{11} + \cdots + 18192720 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 15892705 \beta_{15} + 7986815 \beta_{14} - 33583785 \beta_{13} - 27649920 \beta_{12} + \cdots - 52856019 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\) \(71\)
\(\chi(n)\) \(-1\) \(\beta_{1}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
44.1
2.22190 + 0.111032i
1.51479 − 1.11371i
−2.89591 + 1.43281i
−2.18880 + 2.65755i
0.752308 − 0.673492i
1.45942 + 0.551253i
0.921698 + 0.861704i
0.214591 − 0.363041i
2.22190 − 0.111032i
1.51479 + 1.11371i
−2.89591 − 1.43281i
−2.18880 − 2.65755i
0.752308 + 0.673492i
1.45942 − 0.551253i
0.921698 − 0.861704i
0.214591 + 0.363041i
−1.48938 + 2.57968i 1.18073 + 2.75787i −2.43649 − 4.22013i 3.23955 + 3.80858i −8.87298 − 1.06161i 5.16858 + 4.72078i 2.60040 −6.21175 + 6.51262i −14.6498 + 2.68458i
44.2 −1.48938 + 2.57968i 1.79802 + 2.40148i −2.43649 − 4.22013i −4.91811 − 0.901243i −8.87298 + 1.06161i −5.16858 − 4.72078i 2.60040 −2.53422 + 8.63584i 9.64983 − 11.3448i
44.3 −0.530805 + 0.919382i −1.89916 + 2.32232i 1.43649 + 2.48808i −0.901243 + 4.91811i −1.12702 − 2.97876i 3.04726 − 6.30192i −7.29643 −1.78637 − 8.82093i −4.04323 − 3.43914i
44.4 −0.530805 + 0.919382i 2.96077 − 0.483560i 1.43649 + 2.48808i −3.80858 + 3.23955i −1.12702 + 2.97876i −3.04726 + 6.30192i −7.29643 8.53234 − 2.86342i −0.956769 − 5.22111i
44.5 0.530805 − 0.919382i −2.96077 + 0.483560i 1.43649 + 2.48808i 0.901243 − 4.91811i −1.12702 + 2.97876i 3.04726 − 6.30192i 7.29643 8.53234 − 2.86342i −4.04323 − 3.43914i
44.6 0.530805 − 0.919382i 1.89916 − 2.32232i 1.43649 + 2.48808i 3.80858 − 3.23955i −1.12702 − 2.97876i −3.04726 + 6.30192i 7.29643 −1.78637 − 8.82093i −0.956769 − 5.22111i
44.7 1.48938 − 2.57968i −1.79802 − 2.40148i −2.43649 − 4.22013i −3.23955 − 3.80858i −8.87298 + 1.06161i 5.16858 + 4.72078i −2.60040 −2.53422 + 8.63584i −14.6498 + 2.68458i
44.8 1.48938 − 2.57968i −1.18073 − 2.75787i −2.43649 − 4.22013i 4.91811 + 0.901243i −8.87298 − 1.06161i −5.16858 − 4.72078i −2.60040 −6.21175 + 6.51262i 9.64983 − 11.3448i
74.1 −1.48938 − 2.57968i 1.18073 − 2.75787i −2.43649 + 4.22013i 3.23955 − 3.80858i −8.87298 + 1.06161i 5.16858 − 4.72078i 2.60040 −6.21175 − 6.51262i −14.6498 − 2.68458i
74.2 −1.48938 − 2.57968i 1.79802 − 2.40148i −2.43649 + 4.22013i −4.91811 + 0.901243i −8.87298 − 1.06161i −5.16858 + 4.72078i 2.60040 −2.53422 − 8.63584i 9.64983 + 11.3448i
74.3 −0.530805 − 0.919382i −1.89916 − 2.32232i 1.43649 − 2.48808i −0.901243 − 4.91811i −1.12702 + 2.97876i 3.04726 + 6.30192i −7.29643 −1.78637 + 8.82093i −4.04323 + 3.43914i
74.4 −0.530805 − 0.919382i 2.96077 + 0.483560i 1.43649 − 2.48808i −3.80858 − 3.23955i −1.12702 − 2.97876i −3.04726 − 6.30192i −7.29643 8.53234 + 2.86342i −0.956769 + 5.22111i
74.5 0.530805 + 0.919382i −2.96077 − 0.483560i 1.43649 − 2.48808i 0.901243 + 4.91811i −1.12702 − 2.97876i 3.04726 + 6.30192i 7.29643 8.53234 + 2.86342i −4.04323 + 3.43914i
74.6 0.530805 + 0.919382i 1.89916 + 2.32232i 1.43649 − 2.48808i 3.80858 + 3.23955i −1.12702 + 2.97876i −3.04726 − 6.30192i 7.29643 −1.78637 + 8.82093i −0.956769 + 5.22111i
74.7 1.48938 + 2.57968i −1.79802 + 2.40148i −2.43649 + 4.22013i −3.23955 + 3.80858i −8.87298 − 1.06161i 5.16858 − 4.72078i −2.60040 −2.53422 − 8.63584i −14.6498 − 2.68458i
74.8 1.48938 + 2.57968i −1.18073 + 2.75787i −2.43649 + 4.22013i 4.91811 − 0.901243i −8.87298 + 1.06161i −5.16858 + 4.72078i −2.60040 −6.21175 − 6.51262i 9.64983 + 11.3448i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 44.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.c even 3 1 inner
15.d odd 2 1 inner
21.h odd 6 1 inner
35.j even 6 1 inner
105.o odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.3.o.a ✓ 16
3.b odd 2 1 inner 105.3.o.a ✓ 16
5.b even 2 1 inner 105.3.o.a ✓ 16
7.c even 3 1 inner 105.3.o.a ✓ 16
15.d odd 2 1 inner 105.3.o.a ✓ 16
21.h odd 6 1 inner 105.3.o.a ✓ 16
35.j even 6 1 inner 105.3.o.a ✓ 16
105.o odd 6 1 inner 105.3.o.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.o.a ✓ 16 1.a even 1 1 trivial
105.3.o.a ✓ 16 3.b odd 2 1 inner
105.3.o.a ✓ 16 5.b even 2 1 inner
105.3.o.a ✓ 16 7.c even 3 1 inner
105.3.o.a ✓ 16 15.d odd 2 1 inner
105.3.o.a ✓ 16 21.h odd 6 1 inner
105.3.o.a ✓ 16 35.j even 6 1 inner
105.3.o.a ✓ 16 105.o odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 10T_{2}^{6} + 90T_{2}^{4} + 100T_{2}^{2} + 100 \) acting on \(S_{3}^{\mathrm{new}}(105, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 10 T^{6} + \cdots + 100)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + 4 T^{14} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 152587890625 \) Copy content Toggle raw display
$7$ \( (T^{8} + 52 T^{6} + \cdots + 5764801)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 150 T^{6} + \cdots + 5062500)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 488 T^{2} + 14161)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 250 T^{6} + \cdots + 1464100)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 12 T^{3} + \cdots + 441)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + 1050 T^{6} + \cdots + 63912896100)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 970 T^{2} + 141610)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 86 T^{3} + \cdots + 3200521)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 1694894138161)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 3670 T^{2} + 2199610)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 3272 T^{2} + 2660161)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + 1500 T^{6} + \cdots + 1897473600)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 6452412825600)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 68876886624100)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 98 T^{3} + \cdots + 4104676)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 11943113486161)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 6490 T^{2} + 906010)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} - 10488 T^{6} + \cdots + 1632240801)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 152 T^{3} + \cdots + 31820881)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 20170 T^{2} + 76784410)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 75\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 188 T^{2} + 196)^{4} \) Copy content Toggle raw display
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