Properties

Label 330.2.m.b
Level $330$
Weight $2$
Character orbit 330.m
Analytic conductor $2.635$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 330.m (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.63506326670\)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/330\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(211\) \(221\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−0.309017 0.951057i
0.809017 + 0.587785i
−0.309017 + 0.951057i
0.809017 0.587785i
0.309017 0.951057i 0.809017 0.587785i −0.809017 0.587785i −0.309017 0.951057i −0.309017 0.951057i −1.00000 0.726543i −0.809017 + 0.587785i 0.309017 0.951057i −1.00000
91.1 −0.809017 + 0.587785i −0.309017 0.951057i 0.309017 0.951057i 0.809017 + 0.587785i 0.809017 + 0.587785i −1.00000 + 3.07768i 0.309017 + 0.951057i −0.809017 + 0.587785i −1.00000
181.1 0.309017 + 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i −0.309017 + 0.951057i −0.309017 + 0.951057i −1.00000 + 0.726543i −0.809017 0.587785i 0.309017 + 0.951057i −1.00000
301.1 −0.809017 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i 0.809017 0.587785i 0.809017 0.587785i −1.00000 3.07768i 0.309017 0.951057i −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 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 330.2.m.b 4
3.b odd 2 1 990.2.n.e 4
11.c even 5 1 inner 330.2.m.b 4
11.c even 5 1 3630.2.a.bj 2
11.d odd 10 1 3630.2.a.bb 2
33.h odd 10 1 990.2.n.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.m.b 4 1.a even 1 1 trivial
330.2.m.b 4 11.c even 5 1 inner
990.2.n.e 4 3.b odd 2 1
990.2.n.e 4 33.h odd 10 1
3630.2.a.bb 2 11.d odd 10 1
3630.2.a.bj 2 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 4 T_{7}^{3} + 16 T_{7}^{2} + 24 T_{7} + 16 \) acting on \(S_{2}^{\mathrm{new}}(330, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$3$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$5$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$7$ \( 16 + 24 T + 16 T^{2} + 4 T^{3} + T^{4} \)
$11$ \( 121 - 11 T - 9 T^{2} - T^{3} + T^{4} \)
$13$ \( 361 + 133 T + 24 T^{2} + 2 T^{3} + T^{4} \)
$17$ \( 961 - 372 T + 94 T^{2} - 13 T^{3} + T^{4} \)
$19$ \( 16 - 16 T + 16 T^{2} - 6 T^{3} + T^{4} \)
$23$ \( ( 1 - 7 T + T^{2} )^{2} \)
$29$ \( 1 - 4 T + 6 T^{2} + T^{3} + T^{4} \)
$31$ \( 25 - 25 T + 10 T^{2} + T^{4} \)
$37$ \( 3025 - 275 T + 60 T^{2} - 10 T^{3} + T^{4} \)
$41$ \( 16 - 16 T + 16 T^{2} - 6 T^{3} + T^{4} \)
$43$ \( ( -1 - 11 T + T^{2} )^{2} \)
$47$ \( 2401 + 686 T + 196 T^{2} + 21 T^{3} + T^{4} \)
$53$ \( 16 + 24 T + 76 T^{2} + 14 T^{3} + T^{4} \)
$59$ \( 1681 + 861 T + 166 T^{2} - 4 T^{3} + T^{4} \)
$61$ \( T^{4} \)
$67$ \( ( -31 - T + T^{2} )^{2} \)
$71$ \( 256 - 64 T + 96 T^{2} + 16 T^{3} + T^{4} \)
$73$ \( 32400 - 5400 T + 540 T^{2} - 30 T^{3} + T^{4} \)
$79$ \( 841 - 406 T + 96 T^{2} - 11 T^{3} + T^{4} \)
$83$ \( 256 + 64 T + 16 T^{2} + 4 T^{3} + T^{4} \)
$89$ \( ( 220 - 30 T + T^{2} )^{2} \)
$97$ \( 55696 + 7552 T + 744 T^{2} + 38 T^{3} + T^{4} \)
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