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.

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

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 \(S_{2}^{\mathrm{new}}(30, [\chi])\).