Properties

Label 121.2.c.e
Level $121$
Weight $2$
Character orbit 121.c
Analytic conductor $0.966$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{2} -\zeta_{10}^{2} q^{3} -2 \zeta_{10}^{3} q^{4} -\zeta_{10} q^{5} -2 \zeta_{10} q^{6} + 2 \zeta_{10}^{3} q^{7} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{9} +O(q^{10})\) \( q + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{2} -\zeta_{10}^{2} q^{3} -2 \zeta_{10}^{3} q^{4} -\zeta_{10} q^{5} -2 \zeta_{10} q^{6} + 2 \zeta_{10}^{3} q^{7} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{9} -2 q^{10} -2 q^{12} + ( -4 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{13} + 4 \zeta_{10}^{2} q^{14} + \zeta_{10}^{3} q^{15} + 4 \zeta_{10} q^{16} + 2 \zeta_{10} q^{17} -4 \zeta_{10}^{3} q^{18} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{20} + 2 q^{21} - q^{23} -4 \zeta_{10}^{2} q^{25} + 8 \zeta_{10}^{3} q^{26} -5 \zeta_{10} q^{27} + 4 \zeta_{10} q^{28} + 2 \zeta_{10}^{2} q^{30} + ( -7 + 7 \zeta_{10} - 7 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{31} + 8 q^{32} + 4 q^{34} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{35} -4 \zeta_{10}^{2} q^{36} -3 \zeta_{10}^{3} q^{37} + 4 \zeta_{10} q^{39} -8 \zeta_{10}^{2} q^{41} + ( 4 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{42} -6 q^{43} -2 q^{45} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{46} + 8 \zeta_{10}^{2} q^{47} -4 \zeta_{10}^{3} q^{48} + 3 \zeta_{10} q^{49} -8 \zeta_{10} q^{50} -2 \zeta_{10}^{3} q^{51} + 8 \zeta_{10}^{2} q^{52} + ( 6 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{53} -10 q^{54} -5 \zeta_{10}^{3} q^{59} + 2 \zeta_{10} q^{60} -12 \zeta_{10} q^{61} + 14 \zeta_{10}^{3} q^{62} + 4 \zeta_{10}^{2} q^{63} + ( 8 - 8 \zeta_{10} + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{64} + 4 q^{65} -7 q^{67} + ( 4 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{68} + \zeta_{10}^{2} q^{69} -4 \zeta_{10}^{3} q^{70} + 3 \zeta_{10} q^{71} -4 \zeta_{10}^{3} q^{73} -6 \zeta_{10}^{2} q^{74} + ( -4 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{75} + 8 q^{78} + ( 10 - 10 \zeta_{10} + 10 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{79} -4 \zeta_{10}^{2} q^{80} -\zeta_{10}^{3} q^{81} -16 \zeta_{10} q^{82} + 6 \zeta_{10} q^{83} -4 \zeta_{10}^{3} q^{84} -2 \zeta_{10}^{2} q^{85} + ( -12 + 12 \zeta_{10} - 12 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{86} + 15 q^{89} + ( -4 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{90} -8 \zeta_{10}^{2} q^{91} + 2 \zeta_{10}^{3} q^{92} + 7 \zeta_{10} q^{93} + 16 \zeta_{10} q^{94} -8 \zeta_{10}^{2} q^{96} + ( 7 - 7 \zeta_{10} + 7 \zeta_{10}^{2} - 7 \zeta_{10}^{3} ) q^{97} + 6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} + q^{3} - 2q^{4} - q^{5} - 2q^{6} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{2} + q^{3} - 2q^{4} - q^{5} - 2q^{6} + 2q^{7} + 2q^{9} - 8q^{10} - 8q^{12} - 4q^{13} - 4q^{14} + q^{15} + 4q^{16} + 2q^{17} - 4q^{18} - 2q^{20} + 8q^{21} - 4q^{23} + 4q^{25} + 8q^{26} - 5q^{27} + 4q^{28} - 2q^{30} - 7q^{31} + 32q^{32} + 16q^{34} + 2q^{35} + 4q^{36} - 3q^{37} + 4q^{39} + 8q^{41} + 4q^{42} - 24q^{43} - 8q^{45} - 2q^{46} - 8q^{47} - 4q^{48} + 3q^{49} - 8q^{50} - 2q^{51} - 8q^{52} + 6q^{53} - 40q^{54} - 5q^{59} + 2q^{60} - 12q^{61} + 14q^{62} - 4q^{63} + 8q^{64} + 16q^{65} - 28q^{67} + 4q^{68} - q^{69} - 4q^{70} + 3q^{71} - 4q^{73} + 6q^{74} - 4q^{75} + 32q^{78} + 10q^{79} + 4q^{80} - q^{81} - 16q^{82} + 6q^{83} - 4q^{84} + 2q^{85} - 12q^{86} + 60q^{89} - 4q^{90} + 8q^{91} + 2q^{92} + 7q^{93} + 16q^{94} + 8q^{96} + 7q^{97} + 24q^{98} + O(q^{100}) \)

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
1.61803 1.17557i −0.309017 0.951057i 0.618034 1.90211i −0.809017 0.587785i −1.61803 1.17557i −0.618034 + 1.90211i 0 1.61803 1.17557i −2.00000
9.1 −0.618034 + 1.90211i 0.809017 0.587785i −1.61803 1.17557i 0.309017 + 0.951057i 0.618034 + 1.90211i 1.61803 + 1.17557i 0 −0.618034 + 1.90211i −2.00000
27.1 −0.618034 1.90211i 0.809017 + 0.587785i −1.61803 + 1.17557i 0.309017 0.951057i 0.618034 1.90211i 1.61803 1.17557i 0 −0.618034 1.90211i −2.00000
81.1 1.61803 + 1.17557i −0.309017 + 0.951057i 0.618034 + 1.90211i −0.809017 + 0.587785i −1.61803 + 1.17557i −0.618034 1.90211i 0 1.61803 + 1.17557i −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.2.c.e 4
11.b odd 2 1 121.2.c.a 4
11.c even 5 1 11.2.a.a 1
11.c even 5 3 inner 121.2.c.e 4
11.d odd 10 1 121.2.a.d 1
11.d odd 10 3 121.2.c.a 4
33.f even 10 1 1089.2.a.b 1
33.h odd 10 1 99.2.a.d 1
44.g even 10 1 1936.2.a.i 1
44.h odd 10 1 176.2.a.b 1
55.h odd 10 1 3025.2.a.a 1
55.j even 10 1 275.2.a.b 1
55.k odd 20 2 275.2.b.a 2
77.j odd 10 1 539.2.a.a 1
77.l even 10 1 5929.2.a.h 1
77.m even 15 2 539.2.e.h 2
77.p odd 30 2 539.2.e.g 2
88.k even 10 1 7744.2.a.k 1
88.l odd 10 1 704.2.a.c 1
88.o even 10 1 704.2.a.h 1
88.p odd 10 1 7744.2.a.x 1
99.m even 15 2 891.2.e.k 2
99.n odd 30 2 891.2.e.b 2
132.o even 10 1 1584.2.a.g 1
143.n even 10 1 1859.2.a.b 1
165.o odd 10 1 2475.2.a.a 1
165.v even 20 2 2475.2.c.a 2
176.v odd 20 2 2816.2.c.f 2
176.w even 20 2 2816.2.c.j 2
187.j even 10 1 3179.2.a.a 1
209.m odd 10 1 3971.2.a.b 1
220.n odd 10 1 4400.2.a.i 1
220.v even 20 2 4400.2.b.h 2
231.u even 10 1 4851.2.a.t 1
253.f odd 10 1 5819.2.a.a 1
264.t odd 10 1 6336.2.a.br 1
264.w even 10 1 6336.2.a.bu 1
308.t even 10 1 8624.2.a.j 1
319.k even 10 1 9251.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.2.a.a 1 11.c even 5 1
99.2.a.d 1 33.h odd 10 1
121.2.a.d 1 11.d odd 10 1
121.2.c.a 4 11.b odd 2 1
121.2.c.a 4 11.d odd 10 3
121.2.c.e 4 1.a even 1 1 trivial
121.2.c.e 4 11.c even 5 3 inner
176.2.a.b 1 44.h odd 10 1
275.2.a.b 1 55.j even 10 1
275.2.b.a 2 55.k odd 20 2
539.2.a.a 1 77.j odd 10 1
539.2.e.g 2 77.p odd 30 2
539.2.e.h 2 77.m even 15 2
704.2.a.c 1 88.l odd 10 1
704.2.a.h 1 88.o even 10 1
891.2.e.b 2 99.n odd 30 2
891.2.e.k 2 99.m even 15 2
1089.2.a.b 1 33.f even 10 1
1584.2.a.g 1 132.o even 10 1
1859.2.a.b 1 143.n even 10 1
1936.2.a.i 1 44.g even 10 1
2475.2.a.a 1 165.o odd 10 1
2475.2.c.a 2 165.v even 20 2
2816.2.c.f 2 176.v odd 20 2
2816.2.c.j 2 176.w even 20 2
3025.2.a.a 1 55.h odd 10 1
3179.2.a.a 1 187.j even 10 1
3971.2.a.b 1 209.m odd 10 1
4400.2.a.i 1 220.n odd 10 1
4400.2.b.h 2 220.v even 20 2
4851.2.a.t 1 231.u even 10 1
5819.2.a.a 1 253.f odd 10 1
5929.2.a.h 1 77.l even 10 1
6336.2.a.br 1 264.t odd 10 1
6336.2.a.bu 1 264.w even 10 1
7744.2.a.k 1 88.k even 10 1
7744.2.a.x 1 88.p odd 10 1
8624.2.a.j 1 308.t even 10 1
9251.2.a.d 1 319.k even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2 T_{2}^{3} + 4 T_{2}^{2} - 8 T_{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(121, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 - 8 T + 4 T^{2} - 2 T^{3} + T^{4} \)
$3$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$5$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$7$ \( 16 - 8 T + 4 T^{2} - 2 T^{3} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( 256 + 64 T + 16 T^{2} + 4 T^{3} + T^{4} \)
$17$ \( 16 - 8 T + 4 T^{2} - 2 T^{3} + T^{4} \)
$19$ \( T^{4} \)
$23$ \( ( 1 + T )^{4} \)
$29$ \( T^{4} \)
$31$ \( 2401 + 343 T + 49 T^{2} + 7 T^{3} + T^{4} \)
$37$ \( 81 + 27 T + 9 T^{2} + 3 T^{3} + T^{4} \)
$41$ \( 4096 - 512 T + 64 T^{2} - 8 T^{3} + T^{4} \)
$43$ \( ( 6 + T )^{4} \)
$47$ \( 4096 + 512 T + 64 T^{2} + 8 T^{3} + T^{4} \)
$53$ \( 1296 - 216 T + 36 T^{2} - 6 T^{3} + T^{4} \)
$59$ \( 625 + 125 T + 25 T^{2} + 5 T^{3} + T^{4} \)
$61$ \( 20736 + 1728 T + 144 T^{2} + 12 T^{3} + T^{4} \)
$67$ \( ( 7 + T )^{4} \)
$71$ \( 81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4} \)
$73$ \( 256 + 64 T + 16 T^{2} + 4 T^{3} + T^{4} \)
$79$ \( 10000 - 1000 T + 100 T^{2} - 10 T^{3} + T^{4} \)
$83$ \( 1296 - 216 T + 36 T^{2} - 6 T^{3} + T^{4} \)
$89$ \( ( -15 + T )^{4} \)
$97$ \( 2401 - 343 T + 49 T^{2} - 7 T^{3} + T^{4} \)
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