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Properties

Label 121.5.d.b
Level $121$
Weight $5$
Character orbit 121.d
Analytic conductor $12.508$
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] = [121,5,Mod(40,121)]
 
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([7]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("121.40");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 121.d (of order \(10\), 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: \(12.5077655331\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.5184000000.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 6x^{6} + 36x^{4} - 216x^{2} + 1296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{6} + \cdots + \beta_{4}) q^{2}+ \cdots - 72 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_{6} + \cdots + \beta_{4}) q^{2}+ \cdots + 599 \beta_{6} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{3} + 28 q^{4} - 62 q^{5} + 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{3} + 28 q^{4} - 62 q^{5} + 144 q^{9} + 336 q^{12} - 600 q^{14} + 186 q^{15} + 568 q^{16} + 868 q^{20} + 2216 q^{23} - 672 q^{25} + 2040 q^{26} - 918 q^{27} + 2726 q^{31} + 10080 q^{34} - 2016 q^{36} - 334 q^{37} + 1080 q^{38} + 1800 q^{42} - 17856 q^{45} - 3404 q^{47} - 1704 q^{48} + 1198 q^{49} - 9044 q^{53} + 4800 q^{56} + 13920 q^{58} + 4726 q^{59} - 2604 q^{60} - 6032 q^{64} - 22424 q^{67} + 1662 q^{69} - 18600 q^{70} - 6794 q^{71} + 2016 q^{75} + 24480 q^{78} + 17608 q^{80} - 8910 q^{81} + 11640 q^{82} - 13200 q^{86} - 37384 q^{89} - 20400 q^{91} + 7756 q^{92} - 8178 q^{93} - 8494 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 6x^{6} + 36x^{4} - 216x^{2} + 1296 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} ) / 36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} ) / 216 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 72\nu^{3} + 432\nu ) / 216 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + 3\nu^{3} - 36\nu ) / 18 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 3\nu^{5} + 36\nu^{3} ) / 108 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + 6\nu^{5} + 108\nu ) / 108 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + 2\beta_{6} + 2\beta_{4} ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 6\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 12\beta_{7} + 12\beta_{6} + 6\beta_{5} ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 36\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -36\beta_{6} - 72\beta_{5} - 72\beta_{4} ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 216\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -432\beta_{7} - 432\beta_{5} - 216\beta_{4} ) / 5 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(1 - \beta_{1} + \beta_{2} - \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1
1.43977 + 1.98168i
−1.43977 − 1.98168i
−2.32960 − 0.756934i
2.32960 + 0.756934i
−2.32960 + 0.756934i
2.32960 − 0.756934i
1.43977 − 1.98168i
−1.43977 + 1.98168i
−3.21943 + 4.43117i −0.927051 + 2.85317i −4.32624 − 13.3148i −25.0795 + 18.2213i −9.65830 − 13.2935i −52.0915 + 16.9256i −10.4183 − 3.38511i 58.2492 + 42.3205i − 169.794i
40.2 3.21943 − 4.43117i −0.927051 + 2.85317i −4.32624 − 13.3148i −25.0795 + 18.2213i 9.65830 + 13.2935i 52.0915 − 16.9256i 10.4183 + 3.38511i 58.2492 + 42.3205i 169.794i
94.1 −5.20915 + 1.69256i 2.42705 + 1.76336i 11.3262 − 8.22899i 9.57953 − 29.4828i −15.6275 − 5.07767i 32.1943 + 44.3117i 6.43886 − 8.86234i −22.2492 − 68.4761i 169.794i
94.2 5.20915 − 1.69256i 2.42705 + 1.76336i 11.3262 − 8.22899i 9.57953 − 29.4828i 15.6275 + 5.07767i −32.1943 − 44.3117i −6.43886 + 8.86234i −22.2492 − 68.4761i − 169.794i
112.1 −5.20915 − 1.69256i 2.42705 − 1.76336i 11.3262 + 8.22899i 9.57953 + 29.4828i −15.6275 + 5.07767i 32.1943 − 44.3117i 6.43886 + 8.86234i −22.2492 + 68.4761i − 169.794i
112.2 5.20915 + 1.69256i 2.42705 − 1.76336i 11.3262 + 8.22899i 9.57953 + 29.4828i 15.6275 − 5.07767i −32.1943 + 44.3117i −6.43886 − 8.86234i −22.2492 + 68.4761i 169.794i
118.1 −3.21943 − 4.43117i −0.927051 − 2.85317i −4.32624 + 13.3148i −25.0795 − 18.2213i −9.65830 + 13.2935i −52.0915 − 16.9256i −10.4183 + 3.38511i 58.2492 − 42.3205i 169.794i
118.2 3.21943 + 4.43117i −0.927051 − 2.85317i −4.32624 + 13.3148i −25.0795 − 18.2213i 9.65830 − 13.2935i 52.0915 + 16.9256i 10.4183 − 3.38511i 58.2492 − 42.3205i − 169.794i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.2
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 121.5.d.b 8
11.b odd 2 1 inner 121.5.d.b 8
11.c even 5 1 11.5.b.b ✓ 2
11.c even 5 3 inner 121.5.d.b 8
11.d odd 10 1 11.5.b.b ✓ 2
11.d odd 10 3 inner 121.5.d.b 8
33.f even 10 1 99.5.c.b 2
33.h odd 10 1 99.5.c.b 2
44.g even 10 1 176.5.h.c 2
44.h odd 10 1 176.5.h.c 2
55.h odd 10 1 275.5.c.e 2
55.j even 10 1 275.5.c.e 2
55.k odd 20 2 275.5.d.b 4
55.l even 20 2 275.5.d.b 4
88.k even 10 1 704.5.h.d 2
88.l odd 10 1 704.5.h.d 2
88.o even 10 1 704.5.h.f 2
88.p odd 10 1 704.5.h.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.5.b.b ✓ 2 11.c even 5 1
11.5.b.b ✓ 2 11.d odd 10 1
99.5.c.b 2 33.f even 10 1
99.5.c.b 2 33.h odd 10 1
121.5.d.b 8 1.a even 1 1 trivial
121.5.d.b 8 11.b odd 2 1 inner
121.5.d.b 8 11.c even 5 3 inner
121.5.d.b 8 11.d odd 10 3 inner
176.5.h.c 2 44.g even 10 1
176.5.h.c 2 44.h odd 10 1
275.5.c.e 2 55.h odd 10 1
275.5.c.e 2 55.j even 10 1
275.5.d.b 4 55.k odd 20 2
275.5.d.b 4 55.l even 20 2
704.5.h.d 2 88.k even 10 1
704.5.h.d 2 88.l odd 10 1
704.5.h.f 2 88.o even 10 1
704.5.h.f 2 88.p odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 30T_{2}^{6} + 900T_{2}^{4} - 27000T_{2}^{2} + 810000 \) acting on \(S_{5}^{\mathrm{new}}(121, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 30 T^{6} + \cdots + 810000 \) Copy content Toggle raw display
$3$ \( (T^{4} - 3 T^{3} + 9 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 31 T^{3} + \cdots + 923521)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 81000000000000 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T - 277)^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 3451305657361)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 167 T^{3} + \cdots + 777796321)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{2} + 1452000)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 8391473414416)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 418140497898256)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 31178476245361)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T + 2803)^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 133162575872881)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T + 4673)^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 325333693666081)^{2} \) Copy content Toggle raw display
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