Properties

Label 11.3.d.a
Level 11
Weight 3
Character orbit 11.d
Analytic conductor 0.300
Analytic rank 0
Dimension 4
CM No
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 11 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 11.d (of order \(10\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(0.299728290796\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + ( -2 \zeta_{10} + 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} \) \( + ( -2 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{3} \) \( + ( -4 + 3 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{4} \) \( + 4 \zeta_{10}^{2} q^{5} \) \( + ( 6 - 7 \zeta_{10} + 3 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{6} \) \( + ( 6 - 4 \zeta_{10} + 2 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{7} \) \( + ( 3 + \zeta_{10} + \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{8} \) \( + ( -5 + 5 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -2 \zeta_{10} + 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} \) \( + ( -2 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{3} \) \( + ( -4 + 3 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{4} \) \( + 4 \zeta_{10}^{2} q^{5} \) \( + ( 6 - 7 \zeta_{10} + 3 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{6} \) \( + ( 6 - 4 \zeta_{10} + 2 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{7} \) \( + ( 3 + \zeta_{10} + \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{8} \) \( + ( -5 + 5 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{9} \) \( + ( -4 + 8 \zeta_{10} - 8 \zeta_{10}^{2} ) q^{10} \) \( + ( -3 - 3 \zeta_{10} - 9 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{11} \) \( + ( -2 + 11 \zeta_{10}^{2} - 11 \zeta_{10}^{3} ) q^{12} \) \( + ( -2 - 6 \zeta_{10} + 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{13} \) \( + ( -6 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{14} \) \( + ( 4 - 12 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{15} \) \( + ( 8 \zeta_{10} - 3 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{16} \) \( + ( 13 \zeta_{10} - 13 \zeta_{10}^{3} ) q^{17} \) \( + ( 5 + \zeta_{10} - 7 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{18} \) \( + ( 5 - 2 \zeta_{10} - 2 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{19} \) \( + ( 16 - 16 \zeta_{10} - 4 \zeta_{10}^{3} ) q^{20} \) \( + ( -2 + 4 \zeta_{10} + 4 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{21} \) \( + ( 6 - 16 \zeta_{10} + 29 \zeta_{10}^{2} - 14 \zeta_{10}^{3} ) q^{22} \) \( + ( -4 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{23} \) \( + ( -7 + 9 \zeta_{10} - 16 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{24} \) \( + ( 9 - 9 \zeta_{10} + 9 \zeta_{10}^{2} - 9 \zeta_{10}^{3} ) q^{25} \) \( + ( -10 + 20 \zeta_{10} - 10 \zeta_{10}^{2} ) q^{26} \) \( + ( -2 \zeta_{10} - 19 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{27} \) \( + ( -24 + 2 \zeta_{10} - 12 \zeta_{10}^{2} + 22 \zeta_{10}^{3} ) q^{28} \) \( + ( -18 + 10 \zeta_{10} - 2 \zeta_{10}^{2} + 20 \zeta_{10}^{3} ) q^{29} \) \( + ( -16 + 12 \zeta_{10} + 12 \zeta_{10}^{2} - 16 \zeta_{10}^{3} ) q^{30} \) \( + ( -18 + 18 \zeta_{10} - 4 \zeta_{10}^{3} ) q^{31} \) \( + ( 23 - 46 \zeta_{10} + 22 \zeta_{10}^{2} - 24 \zeta_{10}^{3} ) q^{32} \) \( + ( 3 + 25 \zeta_{10} - 24 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{33} \) \( + ( 13 - 39 \zeta_{10}^{2} + 39 \zeta_{10}^{3} ) q^{34} \) \( + ( 24 + 8 \zeta_{10} + 16 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{35} \) \( + ( 24 - 23 \zeta_{10} + 23 \zeta_{10}^{2} - 24 \zeta_{10}^{3} ) q^{36} \) \( + ( 36 - 18 \zeta_{10} + 36 \zeta_{10}^{2} ) q^{37} \) \( + ( -17 \zeta_{10} + 26 \zeta_{10}^{2} - 17 \zeta_{10}^{3} ) q^{38} \) \( + ( 20 - 20 \zeta_{10} + 10 \zeta_{10}^{2} ) q^{39} \) \( + ( -16 + 4 \zeta_{10} + 8 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{40} \) \( + ( -16 - 27 \zeta_{10} - 27 \zeta_{10}^{2} - 16 \zeta_{10}^{3} ) q^{41} \) \( -10 \zeta_{10}^{3} q^{42} \) \( + ( -18 + 36 \zeta_{10} - 17 \zeta_{10}^{2} + 19 \zeta_{10}^{3} ) q^{43} \) \( + ( -17 + 60 \zeta_{10} - 18 \zeta_{10}^{2} + 14 \zeta_{10}^{3} ) q^{44} \) \( + ( -16 - 20 \zeta_{10}^{2} + 20 \zeta_{10}^{3} ) q^{45} \) \( + ( -2 + 14 \zeta_{10} - 16 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{46} \) \( + ( -26 + 24 \zeta_{10} - 24 \zeta_{10}^{2} + 26 \zeta_{10}^{3} ) q^{47} \) \( + ( -19 + 17 \zeta_{10} - 19 \zeta_{10}^{2} ) q^{48} \) \( + ( -44 \zeta_{10} + 21 \zeta_{10}^{2} - 44 \zeta_{10}^{3} ) q^{49} \) \( + ( -18 + 18 \zeta_{10} - 9 \zeta_{10}^{2} ) q^{50} \) \( + ( -26 - 13 \zeta_{10} + 52 \zeta_{10}^{2} - 26 \zeta_{10}^{3} ) q^{51} \) \( + ( 22 - 4 \zeta_{10} - 4 \zeta_{10}^{2} + 22 \zeta_{10}^{3} ) q^{52} \) \( + ( 30 - 30 \zeta_{10} + 30 \zeta_{10}^{3} ) q^{53} \) \( + ( 17 - 34 \zeta_{10} + 36 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{54} \) \( + ( 8 - 36 \zeta_{10} + 24 \zeta_{10}^{2} - 48 \zeta_{10}^{3} ) q^{55} \) \( + ( 38 + 26 \zeta_{10}^{2} - 26 \zeta_{10}^{3} ) q^{56} \) \( + ( -8 + 26 \zeta_{10} - 34 \zeta_{10}^{2} + 17 \zeta_{10}^{3} ) q^{57} \) \( + ( 12 + 2 \zeta_{10} - 2 \zeta_{10}^{2} - 12 \zeta_{10}^{3} ) q^{58} \) \( + ( 15 - 22 \zeta_{10} + 15 \zeta_{10}^{2} ) q^{59} \) \( + ( 44 \zeta_{10} - 52 \zeta_{10}^{2} + 44 \zeta_{10}^{3} ) q^{60} \) \( + ( 4 + 14 \zeta_{10} + 2 \zeta_{10}^{2} - 18 \zeta_{10}^{3} ) q^{61} \) \( + ( 18 + 22 \zeta_{10} - 62 \zeta_{10}^{2} + 44 \zeta_{10}^{3} ) q^{62} \) \( + ( 18 + 26 \zeta_{10} + 26 \zeta_{10}^{2} + 18 \zeta_{10}^{3} ) q^{63} \) \( + ( -36 + 36 \zeta_{10} - 41 \zeta_{10}^{3} ) q^{64} \) \( + ( -8 + 16 \zeta_{10} - 24 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{65} \) \( + ( 49 - 83 \zeta_{10} + 37 \zeta_{10}^{2} + 14 \zeta_{10}^{3} ) q^{66} \) \( + ( -49 + 17 \zeta_{10}^{2} - 17 \zeta_{10}^{3} ) q^{67} \) \( + ( -13 - 91 \zeta_{10} + 78 \zeta_{10}^{2} - 39 \zeta_{10}^{3} ) q^{68} \) \( + ( 10 - 20 \zeta_{10} + 20 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{69} \) \( + ( -8 - 16 \zeta_{10} - 8 \zeta_{10}^{2} ) q^{70} \) \( + ( 62 \zeta_{10} - 24 \zeta_{10}^{2} + 62 \zeta_{10}^{3} ) q^{71} \) \( + ( -38 + 17 \zeta_{10} - 19 \zeta_{10}^{2} + 21 \zeta_{10}^{3} ) q^{72} \) \( + ( 77 - 17 \zeta_{10} - 43 \zeta_{10}^{2} - 34 \zeta_{10}^{3} ) q^{73} \) \( + ( -54 + 18 \zeta_{10} + 18 \zeta_{10}^{2} - 54 \zeta_{10}^{3} ) q^{74} \) \( + ( 9 - 9 \zeta_{10} + 18 \zeta_{10}^{3} ) q^{75} \) \( + ( -23 + 46 \zeta_{10} - 41 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{76} \) \( + ( -82 + 28 \zeta_{10} - 26 \zeta_{10}^{2} + 74 \zeta_{10}^{3} ) q^{77} \) \( + ( -30 + 40 \zeta_{10}^{2} - 40 \zeta_{10}^{3} ) q^{78} \) \( + ( -12 + 40 \zeta_{10} - 52 \zeta_{10}^{2} + 26 \zeta_{10}^{3} ) q^{79} \) \( + ( -20 - 12 \zeta_{10} + 12 \zeta_{10}^{2} + 20 \zeta_{10}^{3} ) q^{80} \) \( + ( -60 + 64 \zeta_{10} - 60 \zeta_{10}^{2} ) q^{81} \) \( + ( 21 \zeta_{10} + 17 \zeta_{10}^{2} + 21 \zeta_{10}^{3} ) q^{82} \) \( + ( 90 - 47 \zeta_{10} + 45 \zeta_{10}^{2} - 43 \zeta_{10}^{3} ) q^{83} \) \( + ( 32 - 14 \zeta_{10} - 4 \zeta_{10}^{2} - 28 \zeta_{10}^{3} ) q^{84} \) \( + ( 52 + 52 \zeta_{10}^{3} ) q^{85} \) \( + ( 53 - 53 \zeta_{10} + 16 \zeta_{10}^{3} ) q^{86} \) \( + ( 14 - 28 \zeta_{10} + 2 \zeta_{10}^{2} - 26 \zeta_{10}^{3} ) q^{87} \) \( + ( 22 - 44 \zeta_{10} - 22 \zeta_{10}^{2} - 11 \zeta_{10}^{3} ) q^{88} \) \( + ( 72 + 83 \zeta_{10}^{2} - 83 \zeta_{10}^{3} ) q^{89} \) \( + ( 20 - 28 \zeta_{10} + 48 \zeta_{10}^{2} - 24 \zeta_{10}^{3} ) q^{90} \) \( + ( -40 + 20 \zeta_{10} - 20 \zeta_{10}^{2} + 40 \zeta_{10}^{3} ) q^{91} \) \( + ( 22 - 26 \zeta_{10} + 22 \zeta_{10}^{2} ) q^{92} \) \( + ( -58 \zeta_{10} + 84 \zeta_{10}^{2} - 58 \zeta_{10}^{3} ) q^{93} \) \( + ( 48 - 46 \zeta_{10} + 24 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{94} \) \( + ( -12 - 8 \zeta_{10} + 28 \zeta_{10}^{2} - 16 \zeta_{10}^{3} ) q^{95} \) \( + ( 68 - 21 \zeta_{10} - 21 \zeta_{10}^{2} + 68 \zeta_{10}^{3} ) q^{96} \) \( + ( -67 + 67 \zeta_{10} + 36 \zeta_{10}^{3} ) q^{97} \) \( + ( -65 + 130 \zeta_{10} - 86 \zeta_{10}^{2} + 44 \zeta_{10}^{3} ) q^{98} \) \( + ( 28 - 5 \zeta_{10} + 7 \zeta_{10}^{2} - 69 \zeta_{10}^{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 5q^{2} \) \(\mathstrut -\mathstrut 9q^{4} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 15q^{6} \) \(\mathstrut +\mathstrut 10q^{7} \) \(\mathstrut +\mathstrut 15q^{8} \) \(\mathstrut -\mathstrut 11q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 5q^{2} \) \(\mathstrut -\mathstrut 9q^{4} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 15q^{6} \) \(\mathstrut +\mathstrut 10q^{7} \) \(\mathstrut +\mathstrut 15q^{8} \) \(\mathstrut -\mathstrut 11q^{9} \) \(\mathstrut +\mathstrut q^{11} \) \(\mathstrut -\mathstrut 30q^{12} \) \(\mathstrut -\mathstrut 20q^{13} \) \(\mathstrut -\mathstrut 10q^{14} \) \(\mathstrut +\mathstrut 19q^{16} \) \(\mathstrut +\mathstrut 30q^{18} \) \(\mathstrut +\mathstrut 25q^{19} \) \(\mathstrut +\mathstrut 44q^{20} \) \(\mathstrut -\mathstrut 35q^{22} \) \(\mathstrut -\mathstrut 20q^{23} \) \(\mathstrut +\mathstrut 5q^{24} \) \(\mathstrut +\mathstrut 9q^{25} \) \(\mathstrut -\mathstrut 10q^{26} \) \(\mathstrut +\mathstrut 15q^{27} \) \(\mathstrut -\mathstrut 60q^{28} \) \(\mathstrut -\mathstrut 40q^{29} \) \(\mathstrut -\mathstrut 80q^{30} \) \(\mathstrut -\mathstrut 58q^{31} \) \(\mathstrut +\mathstrut 65q^{33} \) \(\mathstrut +\mathstrut 130q^{34} \) \(\mathstrut +\mathstrut 80q^{35} \) \(\mathstrut +\mathstrut 26q^{36} \) \(\mathstrut +\mathstrut 90q^{37} \) \(\mathstrut -\mathstrut 60q^{38} \) \(\mathstrut +\mathstrut 50q^{39} \) \(\mathstrut -\mathstrut 60q^{40} \) \(\mathstrut -\mathstrut 80q^{41} \) \(\mathstrut -\mathstrut 10q^{42} \) \(\mathstrut +\mathstrut 24q^{44} \) \(\mathstrut -\mathstrut 24q^{45} \) \(\mathstrut +\mathstrut 30q^{46} \) \(\mathstrut -\mathstrut 30q^{47} \) \(\mathstrut -\mathstrut 40q^{48} \) \(\mathstrut -\mathstrut 109q^{49} \) \(\mathstrut -\mathstrut 45q^{50} \) \(\mathstrut -\mathstrut 195q^{51} \) \(\mathstrut +\mathstrut 110q^{52} \) \(\mathstrut +\mathstrut 120q^{53} \) \(\mathstrut -\mathstrut 76q^{55} \) \(\mathstrut +\mathstrut 100q^{56} \) \(\mathstrut +\mathstrut 45q^{57} \) \(\mathstrut +\mathstrut 40q^{58} \) \(\mathstrut +\mathstrut 23q^{59} \) \(\mathstrut +\mathstrut 140q^{60} \) \(\mathstrut +\mathstrut 10q^{61} \) \(\mathstrut +\mathstrut 200q^{62} \) \(\mathstrut +\mathstrut 90q^{63} \) \(\mathstrut -\mathstrut 149q^{64} \) \(\mathstrut +\mathstrut 90q^{66} \) \(\mathstrut -\mathstrut 230q^{67} \) \(\mathstrut -\mathstrut 260q^{68} \) \(\mathstrut -\mathstrut 10q^{69} \) \(\mathstrut -\mathstrut 40q^{70} \) \(\mathstrut +\mathstrut 148q^{71} \) \(\mathstrut -\mathstrut 95q^{72} \) \(\mathstrut +\mathstrut 300q^{73} \) \(\mathstrut -\mathstrut 270q^{74} \) \(\mathstrut +\mathstrut 45q^{75} \) \(\mathstrut -\mathstrut 200q^{77} \) \(\mathstrut -\mathstrut 200q^{78} \) \(\mathstrut +\mathstrut 70q^{79} \) \(\mathstrut -\mathstrut 84q^{80} \) \(\mathstrut -\mathstrut 116q^{81} \) \(\mathstrut +\mathstrut 25q^{82} \) \(\mathstrut +\mathstrut 225q^{83} \) \(\mathstrut +\mathstrut 90q^{84} \) \(\mathstrut +\mathstrut 260q^{85} \) \(\mathstrut +\mathstrut 175q^{86} \) \(\mathstrut +\mathstrut 55q^{88} \) \(\mathstrut +\mathstrut 122q^{89} \) \(\mathstrut -\mathstrut 20q^{90} \) \(\mathstrut -\mathstrut 80q^{91} \) \(\mathstrut +\mathstrut 40q^{92} \) \(\mathstrut -\mathstrut 200q^{93} \) \(\mathstrut +\mathstrut 120q^{94} \) \(\mathstrut -\mathstrut 100q^{95} \) \(\mathstrut +\mathstrut 340q^{96} \) \(\mathstrut -\mathstrut 165q^{97} \) \(\mathstrut +\mathstrut 31q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{10}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0.809017 + 0.587785i
0.809017 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
−0.690983 0.224514i −1.11803 + 0.812299i −2.80902 2.04087i 1.23607 + 3.80423i 0.954915 0.310271i 5.85410 8.05748i 3.19098 + 4.39201i −2.19098 + 6.74315i 2.90617i
6.1 −0.690983 + 0.224514i −1.11803 0.812299i −2.80902 + 2.04087i 1.23607 3.80423i 0.954915 + 0.310271i 5.85410 + 8.05748i 3.19098 4.39201i −2.19098 6.74315i 2.90617i
7.1 −1.80902 + 2.48990i 1.11803 3.44095i −1.69098 5.20431i −3.23607 + 2.35114i 6.54508 + 9.00854i −0.854102 + 0.277515i 4.30902 + 1.40008i −3.30902 2.40414i 12.3107i
8.1 −1.80902 2.48990i 1.11803 + 3.44095i −1.69098 + 5.20431i −3.23607 2.35114i 6.54508 9.00854i −0.854102 0.277515i 4.30902 1.40008i −3.30902 + 2.40414i 12.3107i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
11.d Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(11, [\chi])\).