Properties

Label 121.2.c.d
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} + 2 \zeta_{10}^{2} q^{3} + \zeta_{10}^{3} q^{4} -\zeta_{10} q^{5} + 2 \zeta_{10} q^{6} -2 \zeta_{10}^{3} q^{7} + 3 \zeta_{10}^{2} q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} + 2 \zeta_{10}^{2} q^{3} + \zeta_{10}^{3} q^{4} -\zeta_{10} q^{5} + 2 \zeta_{10} q^{6} -2 \zeta_{10}^{3} q^{7} + 3 \zeta_{10}^{2} q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} - q^{10} -2 q^{12} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{13} -2 \zeta_{10}^{2} q^{14} -2 \zeta_{10}^{3} q^{15} + \zeta_{10} q^{16} -5 \zeta_{10} q^{17} + \zeta_{10}^{3} q^{18} -6 \zeta_{10}^{2} q^{19} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{20} + 4 q^{21} + 2 q^{23} + ( -6 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{24} -4 \zeta_{10}^{2} q^{25} -\zeta_{10}^{3} q^{26} + 4 \zeta_{10} q^{27} + 2 \zeta_{10} q^{28} + 9 \zeta_{10}^{3} q^{29} -2 \zeta_{10}^{2} q^{30} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{31} -5 q^{32} -5 q^{34} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{35} -\zeta_{10}^{2} q^{36} + 3 \zeta_{10}^{3} q^{37} -6 \zeta_{10} q^{38} + 2 \zeta_{10} q^{39} -3 \zeta_{10}^{3} q^{40} + 5 \zeta_{10}^{2} q^{41} + ( 4 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{42} + q^{45} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{46} + 2 \zeta_{10}^{2} q^{47} + 2 \zeta_{10}^{3} q^{48} + 3 \zeta_{10} q^{49} -4 \zeta_{10} q^{50} -10 \zeta_{10}^{3} q^{51} + \zeta_{10}^{2} q^{52} + ( -9 + 9 \zeta_{10} - 9 \zeta_{10}^{2} + 9 \zeta_{10}^{3} ) q^{53} + 4 q^{54} + 6 q^{56} + ( 12 - 12 \zeta_{10} + 12 \zeta_{10}^{2} - 12 \zeta_{10}^{3} ) q^{57} + 9 \zeta_{10}^{2} q^{58} -8 \zeta_{10}^{3} q^{59} + 2 \zeta_{10} q^{60} + 6 \zeta_{10} q^{61} -2 \zeta_{10}^{3} q^{62} + 2 \zeta_{10}^{2} q^{63} + ( -7 + 7 \zeta_{10} - 7 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{64} - q^{65} + 2 q^{67} + ( 5 - 5 \zeta_{10} + 5 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{68} + 4 \zeta_{10}^{2} q^{69} + 2 \zeta_{10}^{3} q^{70} -12 \zeta_{10} q^{71} -3 \zeta_{10} q^{72} -2 \zeta_{10}^{3} q^{73} + 3 \zeta_{10}^{2} q^{74} + ( 8 - 8 \zeta_{10} + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{75} + 6 q^{76} + 2 q^{78} + ( -10 + 10 \zeta_{10} - 10 \zeta_{10}^{2} + 10 \zeta_{10}^{3} ) q^{79} -\zeta_{10}^{2} q^{80} + 11 \zeta_{10}^{3} q^{81} + 5 \zeta_{10} q^{82} + 6 \zeta_{10} q^{83} + 4 \zeta_{10}^{3} q^{84} + 5 \zeta_{10}^{2} q^{85} -18 q^{87} -9 q^{89} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{90} -2 \zeta_{10}^{2} q^{91} + 2 \zeta_{10}^{3} q^{92} + 4 \zeta_{10} q^{93} + 2 \zeta_{10} q^{94} + 6 \zeta_{10}^{3} q^{95} -10 \zeta_{10}^{2} q^{96} + ( 13 - 13 \zeta_{10} + 13 \zeta_{10}^{2} - 13 \zeta_{10}^{3} ) q^{97} + 3 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + q^{2} - 2q^{3} + q^{4} - q^{5} + 2q^{6} - 2q^{7} - 3q^{8} - q^{9} + O(q^{10}) \) \( 4q + q^{2} - 2q^{3} + q^{4} - q^{5} + 2q^{6} - 2q^{7} - 3q^{8} - q^{9} - 4q^{10} - 8q^{12} + q^{13} + 2q^{14} - 2q^{15} + q^{16} - 5q^{17} + q^{18} + 6q^{19} + q^{20} + 16q^{21} + 8q^{23} - 6q^{24} + 4q^{25} - q^{26} + 4q^{27} + 2q^{28} + 9q^{29} + 2q^{30} + 2q^{31} - 20q^{32} - 20q^{34} - 2q^{35} + q^{36} + 3q^{37} - 6q^{38} + 2q^{39} - 3q^{40} - 5q^{41} + 4q^{42} + 4q^{45} + 2q^{46} - 2q^{47} + 2q^{48} + 3q^{49} - 4q^{50} - 10q^{51} - q^{52} - 9q^{53} + 16q^{54} + 24q^{56} + 12q^{57} - 9q^{58} - 8q^{59} + 2q^{60} + 6q^{61} - 2q^{62} - 2q^{63} - 7q^{64} - 4q^{65} + 8q^{67} + 5q^{68} - 4q^{69} + 2q^{70} - 12q^{71} - 3q^{72} - 2q^{73} - 3q^{74} + 8q^{75} + 24q^{76} + 8q^{78} - 10q^{79} + q^{80} + 11q^{81} + 5q^{82} + 6q^{83} + 4q^{84} - 5q^{85} - 72q^{87} - 36q^{89} + q^{90} + 2q^{91} + 2q^{92} + 4q^{93} + 2q^{94} + 6q^{95} + 10q^{96} + 13q^{97} + 12q^{98} + O(q^{100}) \)

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
0.809017 0.587785i 0.618034 + 1.90211i −0.309017 + 0.951057i −0.809017 0.587785i 1.61803 + 1.17557i 0.618034 1.90211i 0.927051 + 2.85317i −0.809017 + 0.587785i −1.00000
9.1 −0.309017 + 0.951057i −1.61803 + 1.17557i 0.809017 + 0.587785i 0.309017 + 0.951057i −0.618034 1.90211i −1.61803 1.17557i −2.42705 + 1.76336i 0.309017 0.951057i −1.00000
27.1 −0.309017 0.951057i −1.61803 1.17557i 0.809017 0.587785i 0.309017 0.951057i −0.618034 + 1.90211i −1.61803 + 1.17557i −2.42705 1.76336i 0.309017 + 0.951057i −1.00000
81.1 0.809017 + 0.587785i 0.618034 1.90211i −0.309017 0.951057i −0.809017 + 0.587785i 1.61803 1.17557i 0.618034 + 1.90211i 0.927051 2.85317i −0.809017 0.587785i −1.00000
\(n\): 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.d 4
11.b odd 2 1 121.2.c.b 4
11.c even 5 1 121.2.a.a 1
11.c even 5 3 inner 121.2.c.d 4
11.d odd 10 1 121.2.a.c yes 1
11.d odd 10 3 121.2.c.b 4
33.f even 10 1 1089.2.a.c 1
33.h odd 10 1 1089.2.a.i 1
44.g even 10 1 1936.2.a.b 1
44.h odd 10 1 1936.2.a.a 1
55.h odd 10 1 3025.2.a.b 1
55.j even 10 1 3025.2.a.e 1
77.j odd 10 1 5929.2.a.a 1
77.l even 10 1 5929.2.a.g 1
88.k even 10 1 7744.2.a.bf 1
88.l odd 10 1 7744.2.a.be 1
88.o even 10 1 7744.2.a.f 1
88.p odd 10 1 7744.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.2.a.a 1 11.c even 5 1
121.2.a.c yes 1 11.d odd 10 1
121.2.c.b 4 11.b odd 2 1
121.2.c.b 4 11.d odd 10 3
121.2.c.d 4 1.a even 1 1 trivial
121.2.c.d 4 11.c even 5 3 inner
1089.2.a.c 1 33.f even 10 1
1089.2.a.i 1 33.h odd 10 1
1936.2.a.a 1 44.h odd 10 1
1936.2.a.b 1 44.g even 10 1
3025.2.a.b 1 55.h odd 10 1
3025.2.a.e 1 55.j even 10 1
5929.2.a.a 1 77.j odd 10 1
5929.2.a.g 1 77.l even 10 1
7744.2.a.c 1 88.p odd 10 1
7744.2.a.f 1 88.o even 10 1
7744.2.a.be 1 88.l odd 10 1
7744.2.a.bf 1 88.k even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - T_{2}^{3} + T_{2}^{2} - T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(121, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$3$ \( 16 + 8 T + 4 T^{2} + 2 T^{3} + T^{4} \)
$5$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$7$ \( 16 + 8 T + 4 T^{2} + 2 T^{3} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$17$ \( 625 + 125 T + 25 T^{2} + 5 T^{3} + T^{4} \)
$19$ \( 1296 - 216 T + 36 T^{2} - 6 T^{3} + T^{4} \)
$23$ \( ( -2 + T )^{4} \)
$29$ \( 6561 - 729 T + 81 T^{2} - 9 T^{3} + T^{4} \)
$31$ \( 16 - 8 T + 4 T^{2} - 2 T^{3} + T^{4} \)
$37$ \( 81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4} \)
$41$ \( 625 + 125 T + 25 T^{2} + 5 T^{3} + T^{4} \)
$43$ \( T^{4} \)
$47$ \( 16 + 8 T + 4 T^{2} + 2 T^{3} + T^{4} \)
$53$ \( 6561 + 729 T + 81 T^{2} + 9 T^{3} + T^{4} \)
$59$ \( 4096 + 512 T + 64 T^{2} + 8 T^{3} + T^{4} \)
$61$ \( 1296 - 216 T + 36 T^{2} - 6 T^{3} + T^{4} \)
$67$ \( ( -2 + T )^{4} \)
$71$ \( 20736 + 1728 T + 144 T^{2} + 12 T^{3} + T^{4} \)
$73$ \( 16 + 8 T + 4 T^{2} + 2 T^{3} + T^{4} \)
$79$ \( 10000 + 1000 T + 100 T^{2} + 10 T^{3} + T^{4} \)
$83$ \( 1296 - 216 T + 36 T^{2} - 6 T^{3} + T^{4} \)
$89$ \( ( 9 + T )^{4} \)
$97$ \( 28561 - 2197 T + 169 T^{2} - 13 T^{3} + T^{4} \)
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