Properties

Label 30.2.c.a
Level 30
Weight 2
Character orbit 30.c
Analytic conductor 0.240
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 30.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.239551206064\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/30\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 −2.00000 1.00000i 1.00000 2.00000i 1.00000i −1.00000 −1.00000 + 2.00000i
19.2 1.00000i 1.00000i −1.00000 −2.00000 + 1.00000i 1.00000 2.00000i 1.00000i −1.00000 −1.00000 2.00000i
\(n\): 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
5.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(30, [\chi])\).