Properties

Label 33.2.e.a
Level 33
Weight 2
Character orbit 33.e
Analytic conductor 0.264
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 33 = 3 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 33.e (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.263506326670\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
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.

\(f(q)\) \(=\) \( q + ( -1 - \zeta_{10}^{2} ) q^{2} -\zeta_{10}^{3} q^{3} + ( \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{4} + ( 1 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{5} + ( -1 + \zeta_{10}^{3} ) q^{6} + 3 \zeta_{10}^{2} q^{7} + ( -2 + 2 \zeta_{10} + \zeta_{10}^{3} ) q^{8} -\zeta_{10} q^{9} +O(q^{10})\) \( q + ( -1 - \zeta_{10}^{2} ) q^{2} -\zeta_{10}^{3} q^{3} + ( \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{4} + ( 1 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{5} + ( -1 + \zeta_{10}^{3} ) q^{6} + 3 \zeta_{10}^{2} q^{7} + ( -2 + 2 \zeta_{10} + \zeta_{10}^{3} ) q^{8} -\zeta_{10} q^{9} + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{10} + ( 1 + 3 \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{11} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{12} + ( -2 - 3 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{13} + ( 3 - 3 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{14} + ( \zeta_{10} - 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{15} + ( 3 - 3 \zeta_{10}^{3} ) q^{16} + ( \zeta_{10} - \zeta_{10}^{2} ) q^{17} + ( \zeta_{10} + \zeta_{10}^{3} ) q^{18} + ( -3 + 3 \zeta_{10} - \zeta_{10}^{3} ) q^{19} + ( 2 - 3 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{20} + 3 q^{21} + ( -1 - 2 \zeta_{10} - \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{22} + ( -3 - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{23} + ( 2 - \zeta_{10} + 2 \zeta_{10}^{2} ) q^{24} + ( -3 + 3 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{25} + ( 5 \zeta_{10} + 2 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{26} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{27} + ( -3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{28} + ( -4 \zeta_{10} + 2 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{29} + ( -1 + \zeta_{10} ) q^{30} + ( 3 - \zeta_{10} + 3 \zeta_{10}^{2} ) q^{31} + ( -4 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{32} + ( 2 - 2 \zeta_{10} + 3 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{33} - q^{34} + ( -3 + 6 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{35} + ( 1 - \zeta_{10} ) q^{36} + ( -2 \zeta_{10} - \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{37} + ( 2 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{38} + ( -5 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{39} + ( 3 \zeta_{10} - 4 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{40} + ( 8 - 8 \zeta_{10} - \zeta_{10}^{3} ) q^{41} + ( -3 - 3 \zeta_{10}^{2} ) q^{42} + ( 3 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{43} + ( -3 + 5 \zeta_{10} - 3 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{44} + ( -1 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{45} + ( 3 + 4 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{46} + ( -1 + \zeta_{10} ) q^{47} + ( -3 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{48} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{49} + ( 6 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{50} + ( -\zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{51} + ( 3 - 3 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{52} + ( -1 + 9 \zeta_{10} - \zeta_{10}^{2} ) q^{53} + ( 1 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{54} + ( 5 - 5 \zeta_{10} + 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{55} + ( -3 - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{56} + ( 3 - 4 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{57} + ( -2 + 2 \zeta_{10} + 6 \zeta_{10}^{3} ) q^{58} + ( -7 \zeta_{10} + 6 \zeta_{10}^{2} - 7 \zeta_{10}^{3} ) q^{59} + ( -1 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{60} + ( -3 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{61} + ( -2 \zeta_{10} - 3 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{62} -3 \zeta_{10}^{3} q^{63} + ( -2 - \zeta_{10} - 2 \zeta_{10}^{2} ) q^{64} + ( -3 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{65} + ( -3 - \zeta_{10} - 2 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{66} + ( 5 + 9 \zeta_{10}^{2} - 9 \zeta_{10}^{3} ) q^{67} + ( -1 + 2 \zeta_{10} - \zeta_{10}^{2} ) q^{68} + ( -4 + 4 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{69} + ( -3 \zeta_{10} + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{70} + ( -9 + 9 \zeta_{10}^{3} ) q^{71} + ( 1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{72} + ( 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{73} + ( -3 + 3 \zeta_{10} + 5 \zeta_{10}^{3} ) q^{74} + ( 3 + 3 \zeta_{10}^{2} ) q^{75} + ( -3 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{76} + ( -3 \zeta_{10} + 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{77} + ( 7 + 5 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{78} + ( 4 - 7 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{79} + ( 3 - 3 \zeta_{10} ) q^{80} + \zeta_{10}^{2} q^{81} + ( -9 + 8 \zeta_{10} - 8 \zeta_{10}^{2} + 9 \zeta_{10}^{3} ) q^{82} + ( 9 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 9 \zeta_{10}^{3} ) q^{83} + ( 3 \zeta_{10} - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{84} + ( 2 - 2 \zeta_{10} + \zeta_{10}^{3} ) q^{85} + ( -3 + 2 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{86} + ( -2 - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{87} + ( -6 + 3 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{88} + ( 7 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{89} + ( 1 - \zeta_{10} + \zeta_{10}^{2} ) q^{90} + ( 6 - 6 \zeta_{10} - 15 \zeta_{10}^{3} ) q^{91} + ( \zeta_{10} - 5 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{92} + ( 2 + \zeta_{10} - \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{93} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{94} + ( 7 \zeta_{10} - 11 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{95} + ( 1 - \zeta_{10} + 4 \zeta_{10}^{3} ) q^{96} + ( -13 + 6 \zeta_{10} - 13 \zeta_{10}^{2} ) q^{97} + ( 2 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{98} + ( 1 - 2 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 3q^{2} - q^{3} + 3q^{4} - q^{5} - 3q^{6} - 3q^{7} - 5q^{8} - q^{9} + O(q^{10}) \) \( 4q - 3q^{2} - q^{3} + 3q^{4} - q^{5} - 3q^{6} - 3q^{7} - 5q^{8} - q^{9} + 2q^{10} + 9q^{11} - 2q^{12} - 9q^{13} + 6q^{14} + 4q^{15} + 9q^{16} + 2q^{17} + 2q^{18} - 10q^{19} + 3q^{20} + 12q^{21} - 8q^{22} - 4q^{23} + 5q^{24} - 6q^{25} + 8q^{26} - q^{27} - 6q^{28} - 10q^{29} - 3q^{30} + 8q^{31} - 18q^{32} - q^{33} - 4q^{34} - 3q^{35} + 3q^{36} - 3q^{37} - 9q^{39} + 10q^{40} + 23q^{41} - 9q^{42} + 16q^{43} - 2q^{44} - 6q^{45} + 13q^{46} - 3q^{47} - 6q^{48} - 2q^{49} + 12q^{50} - 3q^{51} + 7q^{52} + 6q^{53} + 2q^{54} + 14q^{55} + 5q^{57} - 20q^{59} + 3q^{60} + 3q^{61} - q^{62} - 3q^{63} - 7q^{64} - 14q^{65} - 8q^{66} + 2q^{67} - q^{68} - 9q^{69} - 9q^{70} - 27q^{71} + 5q^{72} + 6q^{73} - 4q^{74} + 9q^{75} - 20q^{76} - 3q^{77} + 18q^{78} + 5q^{79} + 9q^{80} - q^{81} - 11q^{82} + 21q^{83} + 9q^{84} + 7q^{85} - 7q^{86} - 25q^{88} + 20q^{89} + 2q^{90} + 3q^{91} + 7q^{92} + 8q^{93} + q^{94} + 25q^{95} + 7q^{96} - 33q^{97} + 4q^{98} + 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\) \(13\) \(23\)
\(\chi(n)\) \(\zeta_{10}^{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.809017 + 0.587785i
−0.309017 0.951057i
0.809017 0.587785i
−0.309017 + 0.951057i
−1.30902 0.951057i 0.309017 0.951057i 0.190983 + 0.587785i 0.309017 0.224514i −1.30902 + 0.951057i 0.927051 + 2.85317i −0.690983 + 2.12663i −0.809017 0.587785i −0.618034
16.1 −0.190983 0.587785i −0.809017 0.587785i 1.30902 0.951057i −0.809017 + 2.48990i −0.190983 + 0.587785i −2.42705 + 1.76336i −1.80902 1.31433i 0.309017 + 0.951057i 1.61803
25.1 −1.30902 + 0.951057i 0.309017 + 0.951057i 0.190983 0.587785i 0.309017 + 0.224514i −1.30902 0.951057i 0.927051 2.85317i −0.690983 2.12663i −0.809017 + 0.587785i −0.618034
31.1 −0.190983 + 0.587785i −0.809017 + 0.587785i 1.30902 + 0.951057i −0.809017 2.48990i −0.190983 0.587785i −2.42705 1.76336i −1.80902 + 1.31433i 0.309017 0.951057i 1.61803
\(n\): 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 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.2.e.a 4
3.b odd 2 1 99.2.f.b 4
4.b odd 2 1 528.2.y.f 4
5.b even 2 1 825.2.n.f 4
5.c odd 4 2 825.2.bx.b 8
9.c even 3 2 891.2.n.d 8
9.d odd 6 2 891.2.n.a 8
11.b odd 2 1 363.2.e.j 4
11.c even 5 1 inner 33.2.e.a 4
11.c even 5 1 363.2.a.h 2
11.c even 5 2 363.2.e.h 4
11.d odd 10 1 363.2.a.e 2
11.d odd 10 2 363.2.e.c 4
11.d odd 10 1 363.2.e.j 4
33.f even 10 1 1089.2.a.s 2
33.h odd 10 1 99.2.f.b 4
33.h odd 10 1 1089.2.a.m 2
44.g even 10 1 5808.2.a.bm 2
44.h odd 10 1 528.2.y.f 4
44.h odd 10 1 5808.2.a.bl 2
55.h odd 10 1 9075.2.a.bv 2
55.j even 10 1 825.2.n.f 4
55.j even 10 1 9075.2.a.x 2
55.k odd 20 2 825.2.bx.b 8
99.m even 15 2 891.2.n.d 8
99.n odd 30 2 891.2.n.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.e.a 4 1.a even 1 1 trivial
33.2.e.a 4 11.c even 5 1 inner
99.2.f.b 4 3.b odd 2 1
99.2.f.b 4 33.h odd 10 1
363.2.a.e 2 11.d odd 10 1
363.2.a.h 2 11.c even 5 1
363.2.e.c 4 11.d odd 10 2
363.2.e.h 4 11.c even 5 2
363.2.e.j 4 11.b odd 2 1
363.2.e.j 4 11.d odd 10 1
528.2.y.f 4 4.b odd 2 1
528.2.y.f 4 44.h odd 10 1
825.2.n.f 4 5.b even 2 1
825.2.n.f 4 55.j even 10 1
825.2.bx.b 8 5.c odd 4 2
825.2.bx.b 8 55.k odd 20 2
891.2.n.a 8 9.d odd 6 2
891.2.n.a 8 99.n odd 30 2
891.2.n.d 8 9.c even 3 2
891.2.n.d 8 99.m even 15 2
1089.2.a.m 2 33.h odd 10 1
1089.2.a.s 2 33.f even 10 1
5808.2.a.bl 2 44.h odd 10 1
5808.2.a.bm 2 44.g even 10 1
9075.2.a.x 2 55.j even 10 1
9075.2.a.bv 2 55.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3 T_{2}^{3} + 4 T_{2}^{2} + 2 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(33, [\chi])\).