Properties

Label 121.2.c.c
Level $121$
Weight $2$
Character orbit 121.c
Analytic conductor $0.966$
Analytic rank $0$
Dimension $4$
CM discriminant -11
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 121.c (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.966189864457\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{10}^{2} q^{3} + 2 \zeta_{10}^{3} q^{4} + 3 \zeta_{10} q^{5} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{9} +O(q^{10})\) \( q -\zeta_{10}^{2} q^{3} + 2 \zeta_{10}^{3} q^{4} + 3 \zeta_{10} q^{5} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{9} + 2 q^{12} -3 \zeta_{10}^{3} q^{15} -4 \zeta_{10} q^{16} + ( -6 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{20} -9 q^{23} + 4 \zeta_{10}^{2} q^{25} -5 \zeta_{10} q^{27} + ( 5 - 5 \zeta_{10} + 5 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{31} + 4 \zeta_{10}^{2} q^{36} -7 \zeta_{10}^{3} q^{37} + 6 q^{45} -12 \zeta_{10}^{2} q^{47} + 4 \zeta_{10}^{3} q^{48} + 7 \zeta_{10} q^{49} + ( -6 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{53} + 15 \zeta_{10}^{3} q^{59} + 6 \zeta_{10} q^{60} + ( 8 - 8 \zeta_{10} + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{64} + 13 q^{67} + 9 \zeta_{10}^{2} q^{69} + 3 \zeta_{10} q^{71} + ( 4 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{75} -12 \zeta_{10}^{2} q^{80} -\zeta_{10}^{3} q^{81} -9 q^{89} -18 \zeta_{10}^{3} q^{92} -5 \zeta_{10} q^{93} + ( -17 + 17 \zeta_{10} - 17 \zeta_{10}^{2} + 17 \zeta_{10}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + q^{3} + 2q^{4} + 3q^{5} + 2q^{9} + O(q^{10}) \) \( 4q + q^{3} + 2q^{4} + 3q^{5} + 2q^{9} + 8q^{12} - 3q^{15} - 4q^{16} - 6q^{20} - 36q^{23} - 4q^{25} - 5q^{27} + 5q^{31} - 4q^{36} - 7q^{37} + 24q^{45} + 12q^{47} + 4q^{48} + 7q^{49} - 6q^{53} + 15q^{59} + 6q^{60} + 8q^{64} + 52q^{67} - 9q^{69} + 3q^{71} + 4q^{75} + 12q^{80} - q^{81} - 36q^{89} - 18q^{92} - 5q^{93} - 17q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-\zeta_{10}^{3}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.809017 + 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 0.587785i
0 −0.309017 0.951057i −0.618034 + 1.90211i 2.42705 + 1.76336i 0 0 0 1.61803 1.17557i 0
9.1 0 0.809017 0.587785i 1.61803 + 1.17557i −0.927051 2.85317i 0 0 0 −0.618034 + 1.90211i 0
27.1 0 0.809017 + 0.587785i 1.61803 1.17557i −0.927051 + 2.85317i 0 0 0 −0.618034 1.90211i 0
81.1 0 −0.309017 + 0.951057i −0.618034 1.90211i 2.42705 1.76336i 0 0 0 1.61803 + 1.17557i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
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 121.2.c.c 4
11.b odd 2 1 CM 121.2.c.c 4
11.c even 5 1 121.2.a.b 1
11.c even 5 3 inner 121.2.c.c 4
11.d odd 10 1 121.2.a.b 1
11.d odd 10 3 inner 121.2.c.c 4
33.f even 10 1 1089.2.a.g 1
33.h odd 10 1 1089.2.a.g 1
44.g even 10 1 1936.2.a.h 1
44.h odd 10 1 1936.2.a.h 1
55.h odd 10 1 3025.2.a.d 1
55.j even 10 1 3025.2.a.d 1
77.j odd 10 1 5929.2.a.e 1
77.l even 10 1 5929.2.a.e 1
88.k even 10 1 7744.2.a.n 1
88.l odd 10 1 7744.2.a.n 1
88.o even 10 1 7744.2.a.bb 1
88.p odd 10 1 7744.2.a.bb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.2.a.b 1 11.c even 5 1
121.2.a.b 1 11.d odd 10 1
121.2.c.c 4 1.a even 1 1 trivial
121.2.c.c 4 11.b odd 2 1 CM
121.2.c.c 4 11.c even 5 3 inner
121.2.c.c 4 11.d odd 10 3 inner
1089.2.a.g 1 33.f even 10 1
1089.2.a.g 1 33.h odd 10 1
1936.2.a.h 1 44.g even 10 1
1936.2.a.h 1 44.h odd 10 1
3025.2.a.d 1 55.h odd 10 1
3025.2.a.d 1 55.j even 10 1
5929.2.a.e 1 77.j odd 10 1
5929.2.a.e 1 77.l even 10 1
7744.2.a.n 1 88.k even 10 1
7744.2.a.n 1 88.l odd 10 1
7744.2.a.bb 1 88.o even 10 1
7744.2.a.bb 1 88.p odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(121, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$5$ \( 81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( ( 9 + T )^{4} \)
$29$ \( T^{4} \)
$31$ \( 625 - 125 T + 25 T^{2} - 5 T^{3} + T^{4} \)
$37$ \( 2401 + 343 T + 49 T^{2} + 7 T^{3} + T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( 20736 - 1728 T + 144 T^{2} - 12 T^{3} + T^{4} \)
$53$ \( 1296 + 216 T + 36 T^{2} + 6 T^{3} + T^{4} \)
$59$ \( 50625 - 3375 T + 225 T^{2} - 15 T^{3} + T^{4} \)
$61$ \( T^{4} \)
$67$ \( ( -13 + T )^{4} \)
$71$ \( 81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( ( 9 + T )^{4} \)
$97$ \( 83521 + 4913 T + 289 T^{2} + 17 T^{3} + T^{4} \)
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