Properties

Label 121.4.c.i
Level $121$
Weight $4$
Character orbit 121.c
Analytic conductor $7.139$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [121,4,Mod(3,121)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(121, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("121.3");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 121.c (of order \(5\), degree \(4\), not 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: \(7.13923111069\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.29283765625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 10x^{6} - 19x^{5} + 109x^{4} + 171x^{3} + 810x^{2} + 729x + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_{2} - \beta_1 + 1) q^{2} + ( - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - 2) q^{3} + (\beta_{7} + 3 \beta_{6} - 2 \beta_{5} + 5 \beta_{4} - 2 \beta_1 + 3) q^{4} + ( - 2 \beta_{7} - 2 \beta_{6} + \beta_{5} + 3 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1) q^{5} + (2 \beta_{7} - 13 \beta_{6} + 3 \beta_{5} - 4 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{6} + ( - 5 \beta_{7} + 6 \beta_{6} - 3 \beta_{5} - 3 \beta_1 + 6) q^{7} + (26 \beta_{6} - 3 \beta_{5} + 26 \beta_{4} - 6 \beta_{3} + 7 \beta_{2} - 6 \beta_1 + 7) q^{8} + ( - 14 \beta_{2} + 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_{2} - \beta_1 + 1) q^{2} + ( - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - 2) q^{3} + (\beta_{7} + 3 \beta_{6} - 2 \beta_{5} + 5 \beta_{4} - 2 \beta_1 + 3) q^{4} + ( - 2 \beta_{7} - 2 \beta_{6} + \beta_{5} + 3 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1) q^{5} + (2 \beta_{7} - 13 \beta_{6} + 3 \beta_{5} - 4 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{6} + ( - 5 \beta_{7} + 6 \beta_{6} - 3 \beta_{5} - 3 \beta_1 + 6) q^{7} + (26 \beta_{6} - 3 \beta_{5} + 26 \beta_{4} - 6 \beta_{3} + 7 \beta_{2} - 6 \beta_1 + 7) q^{8} + ( - 14 \beta_{2} + 3 \beta_1) q^{9} + ( - 28 \beta_{6} + 6 \beta_{3} - 28 \beta_{2} - 16) q^{10} + (5 \beta_{7} - 24 \beta_{6} + 5 \beta_{5} - 5 \beta_{4} + 9 \beta_{3} - 24 \beta_{2} + \cdots + 15) q^{12}+ \cdots + (242 \beta_{7} - 1071 \beta_{6} + 242 \beta_{5} - 242 \beta_{4} + \cdots - 843) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 7 q^{2} - 3 q^{3} + 3 q^{4} - 7 q^{5} + 29 q^{6} + 35 q^{7} - 47 q^{8} + 31 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 7 q^{2} - 3 q^{3} + 3 q^{4} - 7 q^{5} + 29 q^{6} + 35 q^{7} - 47 q^{8} + 31 q^{9} - 40 q^{10} + 190 q^{12} + 65 q^{13} - 196 q^{14} - 121 q^{15} - 377 q^{16} + 31 q^{17} + 102 q^{18} - 148 q^{19} + 342 q^{20} - 334 q^{21} - 12 q^{23} + 447 q^{24} - 201 q^{25} - 140 q^{26} + 72 q^{27} + 42 q^{28} + 199 q^{29} + 114 q^{30} - 361 q^{31} - 324 q^{32} - 298 q^{34} - 237 q^{35} + 120 q^{36} + 81 q^{37} - 52 q^{38} - 365 q^{39} - 532 q^{40} + 31 q^{41} + 170 q^{42} + 650 q^{43} + 452 q^{45} - 1204 q^{46} + 857 q^{47} + 644 q^{48} + 1375 q^{49} + 147 q^{50} + 246 q^{51} + 590 q^{52} - 1493 q^{53} + 3100 q^{54} - 1560 q^{56} - 102 q^{57} + 1392 q^{58} + 676 q^{59} + 1068 q^{60} + 525 q^{61} - 2456 q^{62} + 68 q^{63} + 471 q^{64} - 1790 q^{65} + 86 q^{67} - 710 q^{68} - 42 q^{69} - 144 q^{70} + 1143 q^{71} - 919 q^{72} + 2155 q^{73} + 1476 q^{74} - 160 q^{75} + 242 q^{76} - 1340 q^{78} + 861 q^{79} - 1916 q^{80} - 26 q^{81} - 3497 q^{82} - 52 q^{83} + 84 q^{84} - 2383 q^{85} + 1061 q^{86} - 2310 q^{87} + 3782 q^{89} + 1682 q^{90} + 135 q^{91} - 2450 q^{92} - 2077 q^{93} - 702 q^{94} + 1317 q^{95} - 1252 q^{96} - 1344 q^{97} - 2740 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 10x^{6} - 19x^{5} + 109x^{4} + 171x^{3} + 810x^{2} + 729x + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 280\nu ) / 981 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 171 ) / 109 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 1261\nu^{2} ) / 8829 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 10\nu^{6} + 19\nu^{5} - 109\nu^{4} + 1090\nu^{3} - 810\nu^{2} - 729\nu - 6561 ) / 8829 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -19\nu^{7} + 19\nu^{6} - 190\nu^{5} + 1090\nu^{4} - 2071\nu^{3} - 3249\nu^{2} - 15390\nu - 13851 ) / 79461 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10\nu^{7} + 3781\nu^{2} ) / 8829 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + 10\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 9\beta_{6} + 10\beta_{5} + 9\beta_{4} + 9\beta_{2} + 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 19\beta_{7} + 90\beta_{6} + 19\beta_{5} - 19\beta_{4} + 19\beta_{3} + 19\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 109\beta_{3} - 171 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 981\beta_{2} - 280\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1261\beta_{7} - 3781\beta_{4} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
2.86504 2.08157i
−2.05602 + 1.49379i
0.785330 2.41700i
−1.09435 + 3.36805i
0.785330 + 2.41700i
−1.09435 3.36805i
2.86504 + 2.08157i
−2.05602 1.49379i
−0.747004 + 0.542730i −0.476313 1.46594i −2.20868 + 6.79761i −7.05908 5.12872i 1.15142 + 0.836554i −0.239524 + 0.737179i −4.32202 13.3018i 19.9214 14.4737i 8.05666
3.2 4.17405 3.03263i 1.40336 + 4.31911i 5.75376 17.7083i 6.98613 + 5.07572i 18.9560 + 13.7723i −0.513765 + 1.58121i −16.9313 52.1091i 5.15818 3.74763i 44.5532
9.1 −0.903364 + 2.78027i −3.67405 + 2.66936i −0.441690 0.320907i 1.90871 + 5.87440i −4.10252 12.6263i 25.9914 + 18.8838i −17.6291 + 12.8083i −1.97025 + 6.06380i −18.0567
9.2 0.976313 3.00478i 1.24700 0.906001i −1.60339 1.16493i −5.33576 16.4218i −1.50487 4.63152i −7.73807 5.62204i 15.3824 11.1760i −7.60928 + 23.4190i −54.5532
27.1 −0.903364 2.78027i −3.67405 2.66936i −0.441690 + 0.320907i 1.90871 5.87440i −4.10252 + 12.6263i 25.9914 18.8838i −17.6291 12.8083i −1.97025 6.06380i −18.0567
27.2 0.976313 + 3.00478i 1.24700 + 0.906001i −1.60339 + 1.16493i −5.33576 + 16.4218i −1.50487 + 4.63152i −7.73807 + 5.62204i 15.3824 + 11.1760i −7.60928 23.4190i −54.5532
81.1 −0.747004 0.542730i −0.476313 + 1.46594i −2.20868 6.79761i −7.05908 + 5.12872i 1.15142 0.836554i −0.239524 0.737179i −4.32202 + 13.3018i 19.9214 + 14.4737i 8.05666
81.2 4.17405 + 3.03263i 1.40336 4.31911i 5.75376 + 17.7083i 6.98613 5.07572i 18.9560 13.7723i −0.513765 1.58121i −16.9313 + 52.1091i 5.15818 + 3.74763i 44.5532
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.4.c.i 8
11.b odd 2 1 11.4.c.a 8
11.c even 5 1 121.4.a.f 4
11.c even 5 2 121.4.c.b 8
11.c even 5 1 inner 121.4.c.i 8
11.d odd 10 1 11.4.c.a 8
11.d odd 10 1 121.4.a.g 4
11.d odd 10 2 121.4.c.h 8
33.d even 2 1 99.4.f.c 8
33.f even 10 1 99.4.f.c 8
33.f even 10 1 1089.4.a.y 4
33.h odd 10 1 1089.4.a.bh 4
44.c even 2 1 176.4.m.c 8
44.g even 10 1 176.4.m.c 8
44.g even 10 1 1936.4.a.bk 4
44.h odd 10 1 1936.4.a.bl 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.c.a 8 11.b odd 2 1
11.4.c.a 8 11.d odd 10 1
99.4.f.c 8 33.d even 2 1
99.4.f.c 8 33.f even 10 1
121.4.a.f 4 11.c even 5 1
121.4.a.g 4 11.d odd 10 1
121.4.c.b 8 11.c even 5 2
121.4.c.h 8 11.d odd 10 2
121.4.c.i 8 1.a even 1 1 trivial
121.4.c.i 8 11.c even 5 1 inner
176.4.m.c 8 44.c even 2 1
176.4.m.c 8 44.g even 10 1
1089.4.a.y 4 33.f even 10 1
1089.4.a.bh 4 33.h odd 10 1
1936.4.a.bk 4 44.g even 10 1
1936.4.a.bl 4 44.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 7T_{2}^{7} + 31T_{2}^{6} - 71T_{2}^{5} + 319T_{2}^{4} - 78T_{2}^{3} + 1664T_{2}^{2} + 2816T_{2} + 1936 \) acting on \(S_{4}^{\mathrm{new}}(121, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 7 T^{7} + 31 T^{6} + \cdots + 1936 \) Copy content Toggle raw display
$3$ \( T^{8} + 3 T^{7} + 16 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$5$ \( T^{8} + 7 T^{7} + 250 T^{6} + \cdots + 64577296 \) Copy content Toggle raw display
$7$ \( T^{8} - 35 T^{7} + 268 T^{6} + \cdots + 156816 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 65 T^{7} + \cdots + 2905210000 \) Copy content Toggle raw display
$17$ \( T^{8} - 31 T^{7} + \cdots + 9608691844521 \) Copy content Toggle raw display
$19$ \( T^{8} + 148 T^{7} + \cdots + 90104309844241 \) Copy content Toggle raw display
$23$ \( (T^{4} + 6 T^{3} - 21180 T^{2} + \cdots + 49883584)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 199 T^{7} + \cdots + 97\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( T^{8} + 361 T^{7} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{8} - 81 T^{7} + \cdots + 26\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{8} - 31 T^{7} + \cdots + 27\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( (T^{4} - 325 T^{3} - 77011 T^{2} + \cdots - 1288748736)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 857 T^{7} + \cdots + 70\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{8} + 1493 T^{7} + \cdots + 71\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{8} - 676 T^{7} + \cdots + 66\!\cdots\!61 \) Copy content Toggle raw display
$61$ \( T^{8} - 525 T^{7} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( (T^{4} - 43 T^{3} - 394721 T^{2} + \cdots - 8869996224)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 1143 T^{7} + \cdots + 55\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{8} - 2155 T^{7} + \cdots + 14\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( T^{8} - 861 T^{7} + \cdots + 44\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{8} + 52 T^{7} + \cdots + 32\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( (T^{4} - 1891 T^{3} + \cdots - 2046678844)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 1344 T^{7} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
show more
show less