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Properties

Label 121.4.c.b
Level $121$
Weight $4$
Character orbit 121.c
Analytic conductor $7.139$
Analytic rank $0$
Dimension $8$
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] = [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.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_{5} + \beta_{4}) q^{2} + (\beta_{7} + \beta_{6} + \beta_{5} + \cdots + \beta_1) q^{3}+ \cdots + (17 \beta_{6} - 3 \beta_{5} + \cdots + 17) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_{5} + \beta_{4}) q^{2} + (\beta_{7} + \beta_{6} + \beta_{5} + \cdots + \beta_1) q^{3}+ \cdots + ( - 242 \beta_{7} - 829 \beta_{6} + \cdots - 792) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3 q^{2} - 3 q^{3} - 17 q^{4} + 18 q^{5} + 14 q^{6} - 10 q^{7} + 113 q^{8} + 31 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 3 q^{2} - 3 q^{3} - 17 q^{4} + 18 q^{5} + 14 q^{6} - 10 q^{7} + 113 q^{8} + 31 q^{9} - 40 q^{10} + 190 q^{12} - 40 q^{13} + 64 q^{14} + 119 q^{15} + 343 q^{16} + 201 q^{17} - 53 q^{18} + 302 q^{19} - 88 q^{20} - 334 q^{21} - 12 q^{23} - 348 q^{24} + 214 q^{25} + 340 q^{26} + 72 q^{27} - 18 q^{28} + 164 q^{29} - 306 q^{30} + 324 q^{31} - 324 q^{32} - 298 q^{34} + 593 q^{35} - 125 q^{36} - 174 q^{37} - 327 q^{38} + 125 q^{39} - 282 q^{40} + 121 q^{41} - 590 q^{42} + 650 q^{43} + 452 q^{45} - 54 q^{46} + 12 q^{47} - 991 q^{48} - 1130 q^{49} - 1163 q^{50} - 194 q^{51} - 1600 q^{52} + 682 q^{53} + 3100 q^{54} - 1560 q^{56} + 333 q^{57} - 2348 q^{58} - 854 q^{59} - 502 q^{60} - 630 q^{61} + 2114 q^{62} - 67 q^{63} - 1289 q^{64} - 1790 q^{65} + 86 q^{67} + 575 q^{68} + 1198 q^{69} + 166 q^{70} - 1772 q^{71} - 104 q^{72} - 2425 q^{73} - 1274 q^{74} + 975 q^{75} + 242 q^{76} - 1340 q^{78} + 116 q^{79} + 1794 q^{80} - 26 q^{81} + 2708 q^{82} + 1738 q^{83} + 174 q^{84} + 1662 q^{85} - 784 q^{86} - 2310 q^{87} + 3782 q^{89} - 2658 q^{90} - 55 q^{91} + 5900 q^{92} + 2743 q^{93} + 1458 q^{94} + 487 q^{95} + 2383 q^{96} + 3116 q^{97} - 2740 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

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

\(\nu\)\(=\) \( \beta_1 \) 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

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

Character values

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

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

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
0.785330 − 2.41700i
−1.09435 + 3.36805i
2.86504 + 2.08157i
−2.05602 − 1.49379i
2.86504 − 2.08157i
−2.05602 + 1.49379i
0.785330 + 2.41700i
−1.09435 − 3.36805i
−2.55602 + 1.85706i −0.476313 − 1.46594i 0.612442 − 1.88490i 13.9692 + 10.1492i 3.93980 + 2.86243i 2.95568 − 9.09665i −5.87554 − 18.0831i 19.9214 − 14.4737i −54.5532
3.2 2.36504 − 1.71830i 1.40336 + 4.31911i 0.168711 − 0.519238i −4.99706 − 3.63058i 10.7405 + 7.80346i −9.92782 + 30.5547i 6.73371 + 20.7242i 5.15818 − 3.74763i −18.0567
9.1 −1.59435 + 4.90690i −3.67405 + 2.66936i −15.0635 − 10.9443i −2.66846 − 8.21268i −7.24054 − 22.2841i 1.34505 + 0.977239i 44.3266 − 32.2052i −1.97025 + 6.06380i 44.5532
9.2 0.285330 − 0.878155i 1.24700 − 0.906001i 5.78239 + 4.20115i 2.69633 + 8.29844i −0.439802 − 1.35357i 0.627082 + 0.455601i 11.3152 − 8.22096i −7.60928 + 23.4190i 8.05666
27.1 −1.59435 − 4.90690i −3.67405 − 2.66936i −15.0635 + 10.9443i −2.66846 + 8.21268i −7.24054 + 22.2841i 1.34505 − 0.977239i 44.3266 + 32.2052i −1.97025 − 6.06380i 44.5532
27.2 0.285330 + 0.878155i 1.24700 + 0.906001i 5.78239 − 4.20115i 2.69633 − 8.29844i −0.439802 + 1.35357i 0.627082 − 0.455601i 11.3152 + 8.22096i −7.60928 − 23.4190i 8.05666
81.1 −2.55602 − 1.85706i −0.476313 + 1.46594i 0.612442 + 1.88490i 13.9692 − 10.1492i 3.93980 − 2.86243i 2.95568 + 9.09665i −5.87554 + 18.0831i 19.9214 + 14.4737i −54.5532
81.2 2.36504 + 1.71830i 1.40336 − 4.31911i 0.168711 + 0.519238i −4.99706 + 3.63058i 10.7405 − 7.80346i −9.92782 − 30.5547i 6.73371 − 20.7242i 5.15818 + 3.74763i −18.0567
\(n\): 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.b 8
11.b odd 2 1 121.4.c.h 8
11.c even 5 1 121.4.a.f 4
11.c even 5 1 inner 121.4.c.b 8
11.c even 5 2 121.4.c.i 8
11.d odd 10 2 11.4.c.a ✓ 8
11.d odd 10 1 121.4.a.g 4
11.d odd 10 1 121.4.c.h 8
33.f even 10 2 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.g even 10 2 176.4.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.d odd 10 2
99.4.f.c 8 33.f even 10 2
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 1.a even 1 1 trivial
121.4.c.b 8 11.c even 5 1 inner
121.4.c.h 8 11.b odd 2 1
121.4.c.h 8 11.d odd 10 1
121.4.c.i 8 11.c even 5 2
176.4.m.c 8 44.g even 10 2
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 \( T_{2}^{8} + 3T_{2}^{7} + 21T_{2}^{6} - 21T_{2}^{5} - 51T_{2}^{4} + 212T_{2}^{3} + 2104T_{2}^{2} - 1144T_{2} + 1936 \) acting on \(S_{4}^{\mathrm{new}}(121, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 3 T^{7} + \cdots + 1936 \) Copy content Toggle raw display
$3$ \( T^{8} + 3 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$5$ \( T^{8} - 18 T^{7} + \cdots + 64577296 \) Copy content Toggle raw display
$7$ \( T^{8} + 10 T^{7} + \cdots + 156816 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 2905210000 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 9608691844521 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 90104309844241 \) Copy content Toggle raw display
$23$ \( (T^{4} + 6 T^{3} + \cdots + 49883584)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 97\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 26\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 27\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( (T^{4} - 325 T^{3} + \cdots - 1288748736)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 70\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 71\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 66\!\cdots\!61 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( (T^{4} - 43 T^{3} + \cdots - 8869996224)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 55\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 14\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 44\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 32\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( (T^{4} - 1891 T^{3} + \cdots - 2046678844)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
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