Properties

Label 11.5.b.b
Level 11
Weight 5
Character orbit 11.b
Analytic conductor 1.137
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 11 \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 11.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.13706959392\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-30}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-30}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( -3 q^{3} \) \( -14 q^{4} \) \( + 31 q^{5} \) \( -3 \beta q^{6} \) \( -10 \beta q^{7} \) \( + 2 \beta q^{8} \) \( -72 q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( -3 q^{3} \) \( -14 q^{4} \) \( + 31 q^{5} \) \( -3 \beta q^{6} \) \( -10 \beta q^{7} \) \( + 2 \beta q^{8} \) \( -72 q^{9} \) \( + 31 \beta q^{10} \) \( + ( 11 - 22 \beta ) q^{11} \) \( + 42 q^{12} \) \( + 34 \beta q^{13} \) \( + 300 q^{14} \) \( -93 q^{15} \) \( -284 q^{16} \) \( -42 \beta q^{17} \) \( -72 \beta q^{18} \) \( + 18 \beta q^{19} \) \( -434 q^{20} \) \( + 30 \beta q^{21} \) \( + ( 660 + 11 \beta ) q^{22} \) \( + 277 q^{23} \) \( -6 \beta q^{24} \) \( + 336 q^{25} \) \( -1020 q^{26} \) \( + 459 q^{27} \) \( + 140 \beta q^{28} \) \( + 232 \beta q^{29} \) \( -93 \beta q^{30} \) \( -1363 q^{31} \) \( -252 \beta q^{32} \) \( + ( -33 + 66 \beta ) q^{33} \) \( + 1260 q^{34} \) \( -310 \beta q^{35} \) \( + 1008 q^{36} \) \( + 167 q^{37} \) \( -540 q^{38} \) \( -102 \beta q^{39} \) \( + 62 \beta q^{40} \) \( + 194 \beta q^{41} \) \( -900 q^{42} \) \( -220 \beta q^{43} \) \( + ( -154 + 308 \beta ) q^{44} \) \( -2232 q^{45} \) \( + 277 \beta q^{46} \) \( + 1702 q^{47} \) \( + 852 q^{48} \) \( -599 q^{49} \) \( + 336 \beta q^{50} \) \( + 126 \beta q^{51} \) \( -476 \beta q^{52} \) \( + 4522 q^{53} \) \( + 459 \beta q^{54} \) \( + ( 341 - 682 \beta ) q^{55} \) \( + 600 q^{56} \) \( -54 \beta q^{57} \) \( -6960 q^{58} \) \( -2363 q^{59} \) \( + 1302 q^{60} \) \( -724 \beta q^{61} \) \( -1363 \beta q^{62} \) \( + 720 \beta q^{63} \) \( + 3016 q^{64} \) \( + 1054 \beta q^{65} \) \( + ( -1980 - 33 \beta ) q^{66} \) \( -2803 q^{67} \) \( + 588 \beta q^{68} \) \( -831 q^{69} \) \( + 9300 q^{70} \) \( + 3397 q^{71} \) \( -144 \beta q^{72} \) \( + 606 \beta q^{73} \) \( + 167 \beta q^{74} \) \( -1008 q^{75} \) \( -252 \beta q^{76} \) \( + ( -6600 - 110 \beta ) q^{77} \) \( + 3060 q^{78} \) \( + 1112 \beta q^{79} \) \( -8804 q^{80} \) \( + 4455 q^{81} \) \( -5820 q^{82} \) \( -152 \beta q^{83} \) \( -420 \beta q^{84} \) \( -1302 \beta q^{85} \) \( + 6600 q^{86} \) \( -696 \beta q^{87} \) \( + ( 1320 + 22 \beta ) q^{88} \) \( -4673 q^{89} \) \( -2232 \beta q^{90} \) \( + 10200 q^{91} \) \( -3878 q^{92} \) \( + 4089 q^{93} \) \( + 1702 \beta q^{94} \) \( + 558 \beta q^{95} \) \( + 756 \beta q^{96} \) \( + 4247 q^{97} \) \( -599 \beta q^{98} \) \( + ( -792 + 1584 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 28q^{4} \) \(\mathstrut +\mathstrut 62q^{5} \) \(\mathstrut -\mathstrut 144q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 28q^{4} \) \(\mathstrut +\mathstrut 62q^{5} \) \(\mathstrut -\mathstrut 144q^{9} \) \(\mathstrut +\mathstrut 22q^{11} \) \(\mathstrut +\mathstrut 84q^{12} \) \(\mathstrut +\mathstrut 600q^{14} \) \(\mathstrut -\mathstrut 186q^{15} \) \(\mathstrut -\mathstrut 568q^{16} \) \(\mathstrut -\mathstrut 868q^{20} \) \(\mathstrut +\mathstrut 1320q^{22} \) \(\mathstrut +\mathstrut 554q^{23} \) \(\mathstrut +\mathstrut 672q^{25} \) \(\mathstrut -\mathstrut 2040q^{26} \) \(\mathstrut +\mathstrut 918q^{27} \) \(\mathstrut -\mathstrut 2726q^{31} \) \(\mathstrut -\mathstrut 66q^{33} \) \(\mathstrut +\mathstrut 2520q^{34} \) \(\mathstrut +\mathstrut 2016q^{36} \) \(\mathstrut +\mathstrut 334q^{37} \) \(\mathstrut -\mathstrut 1080q^{38} \) \(\mathstrut -\mathstrut 1800q^{42} \) \(\mathstrut -\mathstrut 308q^{44} \) \(\mathstrut -\mathstrut 4464q^{45} \) \(\mathstrut +\mathstrut 3404q^{47} \) \(\mathstrut +\mathstrut 1704q^{48} \) \(\mathstrut -\mathstrut 1198q^{49} \) \(\mathstrut +\mathstrut 9044q^{53} \) \(\mathstrut +\mathstrut 682q^{55} \) \(\mathstrut +\mathstrut 1200q^{56} \) \(\mathstrut -\mathstrut 13920q^{58} \) \(\mathstrut -\mathstrut 4726q^{59} \) \(\mathstrut +\mathstrut 2604q^{60} \) \(\mathstrut +\mathstrut 6032q^{64} \) \(\mathstrut -\mathstrut 3960q^{66} \) \(\mathstrut -\mathstrut 5606q^{67} \) \(\mathstrut -\mathstrut 1662q^{69} \) \(\mathstrut +\mathstrut 18600q^{70} \) \(\mathstrut +\mathstrut 6794q^{71} \) \(\mathstrut -\mathstrut 2016q^{75} \) \(\mathstrut -\mathstrut 13200q^{77} \) \(\mathstrut +\mathstrut 6120q^{78} \) \(\mathstrut -\mathstrut 17608q^{80} \) \(\mathstrut +\mathstrut 8910q^{81} \) \(\mathstrut -\mathstrut 11640q^{82} \) \(\mathstrut +\mathstrut 13200q^{86} \) \(\mathstrut +\mathstrut 2640q^{88} \) \(\mathstrut -\mathstrut 9346q^{89} \) \(\mathstrut +\mathstrut 20400q^{91} \) \(\mathstrut -\mathstrut 7756q^{92} \) \(\mathstrut +\mathstrut 8178q^{93} \) \(\mathstrut +\mathstrut 8494q^{97} \) \(\mathstrut -\mathstrut 1584q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
5.47723i
5.47723i
5.47723i −3.00000 −14.0000 31.0000 16.4317i 54.7723i 10.9545i −72.0000 169.794i
10.2 5.47723i −3.00000 −14.0000 31.0000 16.4317i 54.7723i 10.9545i −72.0000 169.794i
\(n\): 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.b Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut +\mathstrut 30 \) acting on \(S_{5}^{\mathrm{new}}(11, [\chi])\).