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Properties

Label 121.4.c.e
Level $121$
Weight $4$
Character orbit 121.c
Analytic conductor $7.139$
Analytic rank $0$
Dimension $8$
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] = [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: 8.0.1827904000000.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 26x^{6} + 676x^{4} + 17576x^{2} + 456976 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} - 5 \beta_{6} q^{3} + 18 \beta_{4} q^{4} + ( - 5 \beta_{6} - 5 \beta_{4} + \cdots - 5) q^{5}+ \cdots - 2 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} - 5 \beta_{6} q^{3} + 18 \beta_{4} q^{4} + ( - 5 \beta_{6} - 5 \beta_{4} + \cdots - 5) q^{5}+ \cdots + 73 \beta_{5} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 10 q^{3} - 36 q^{4} - 10 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 10 q^{3} - 36 q^{4} - 10 q^{5} + 4 q^{9} - 720 q^{12} - 208 q^{14} + 50 q^{15} - 232 q^{16} - 180 q^{20} + 280 q^{23} + 200 q^{25} - 624 q^{26} - 290 q^{27} - 30 q^{31} - 832 q^{34} + 72 q^{36} + 530 q^{37} + 1040 q^{38} + 1040 q^{42} - 80 q^{45} - 760 q^{47} + 1160 q^{48} - 146 q^{49} - 1020 q^{53} + 8320 q^{56} + 2080 q^{58} - 42 q^{59} + 900 q^{60} - 16 q^{64} + 4680 q^{67} + 350 q^{69} - 1040 q^{70} - 626 q^{71} - 1000 q^{75} - 12480 q^{78} - 1160 q^{80} + 1342 q^{81} + 1040 q^{82} + 4576 q^{86} - 1480 q^{89} - 2496 q^{91} - 1260 q^{92} + 150 q^{93} - 1570 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 26x^{6} + 676x^{4} + 17576x^{2} + 456976 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 26 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 26 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 676 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 676 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 17576 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 17576 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 26\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 26\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 676\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 676\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 17576\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 17576\beta_{7} \) Copy content Toggle raw display

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

Character values

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

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

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
1.57568 + 4.84946i
−1.57568 − 4.84946i
4.12519 − 2.99713i
−4.12519 + 2.99713i
4.12519 + 2.99713i
−4.12519 − 2.99713i
1.57568 − 4.84946i
−1.57568 + 4.84946i
−4.12519 + 2.99713i −1.54508 − 4.75528i 5.56231 − 17.1190i −4.04508 − 2.93893i 20.6260 + 14.9856i 6.30273 − 19.3978i 15.7568 + 48.4946i 1.61803 − 1.17557i 25.4951
3.2 4.12519 − 2.99713i −1.54508 − 4.75528i 5.56231 − 17.1190i −4.04508 − 2.93893i −20.6260 − 14.9856i −6.30273 + 19.3978i −15.7568 − 48.4946i 1.61803 − 1.17557i −25.4951
9.1 −1.57568 + 4.84946i 4.04508 − 2.93893i −14.5623 − 10.5801i 1.54508 + 4.75528i 7.87842 + 24.2473i 16.5008 + 11.9885i 41.2519 − 29.9713i −0.618034 + 1.90211i −25.4951
9.2 1.57568 − 4.84946i 4.04508 − 2.93893i −14.5623 − 10.5801i 1.54508 + 4.75528i −7.87842 − 24.2473i −16.5008 − 11.9885i −41.2519 + 29.9713i −0.618034 + 1.90211i 25.4951
27.1 −1.57568 − 4.84946i 4.04508 + 2.93893i −14.5623 + 10.5801i 1.54508 − 4.75528i 7.87842 − 24.2473i 16.5008 − 11.9885i 41.2519 + 29.9713i −0.618034 − 1.90211i −25.4951
27.2 1.57568 + 4.84946i 4.04508 + 2.93893i −14.5623 + 10.5801i 1.54508 − 4.75528i −7.87842 + 24.2473i −16.5008 + 11.9885i −41.2519 − 29.9713i −0.618034 − 1.90211i 25.4951
81.1 −4.12519 − 2.99713i −1.54508 + 4.75528i 5.56231 + 17.1190i −4.04508 + 2.93893i 20.6260 − 14.9856i 6.30273 + 19.3978i 15.7568 − 48.4946i 1.61803 + 1.17557i 25.4951
81.2 4.12519 + 2.99713i −1.54508 + 4.75528i 5.56231 + 17.1190i −4.04508 + 2.93893i −20.6260 + 14.9856i −6.30273 − 19.3978i −15.7568 + 48.4946i 1.61803 + 1.17557i −25.4951
\(n\): 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.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.4.c.e 8
11.b odd 2 1 inner 121.4.c.e 8
11.c even 5 1 121.4.a.d ✓ 2
11.c even 5 3 inner 121.4.c.e 8
11.d odd 10 1 121.4.a.d ✓ 2
11.d odd 10 3 inner 121.4.c.e 8
33.f even 10 1 1089.4.a.r 2
33.h odd 10 1 1089.4.a.r 2
44.g even 10 1 1936.4.a.ba 2
44.h odd 10 1 1936.4.a.ba 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.4.a.d ✓ 2 11.c even 5 1
121.4.a.d ✓ 2 11.d odd 10 1
121.4.c.e 8 1.a even 1 1 trivial
121.4.c.e 8 11.b odd 2 1 inner
121.4.c.e 8 11.c even 5 3 inner
121.4.c.e 8 11.d odd 10 3 inner
1089.4.a.r 2 33.f even 10 1
1089.4.a.r 2 33.h odd 10 1
1936.4.a.ba 2 44.g even 10 1
1936.4.a.ba 2 44.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 26T_{2}^{6} + 676T_{2}^{4} + 17576T_{2}^{2} + 456976 \) acting on \(S_{4}^{\mathrm{new}}(121, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 26 T^{6} + \cdots + 456976 \) Copy content Toggle raw display
$3$ \( (T^{4} - 5 T^{3} + \cdots + 625)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 5 T^{3} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 29948379136 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 196491315511296 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 29948379136 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T - 35)^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + 15 T^{3} + \cdots + 50625)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 265 T^{3} + \cdots + 4931550625)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{2} - 201344)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 380 T^{3} + \cdots + 20851360000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 510 T^{3} + \cdots + 67652010000)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 21 T^{3} + \cdots + 194481)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T - 585)^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} + 313 T^{3} + \cdots + 9597924961)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 32\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T + 185)^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} + 785 T^{3} + \cdots + 379733250625)^{2} \) Copy content Toggle raw display
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