Properties

Label 30.2.e.a
Level 30
Weight 2
Character orbit 30.e
Analytic conductor 0.240
Analytic rank 0
Dimension 4
CM No
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 30.e (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.239551206064\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

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

Character Values

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

\(n\) \(7\) \(11\)
\(\chi(n)\) \(-\zeta_{8}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i −0.292893 + 1.70711i 1.00000i 0.707107 2.12132i −1.00000 1.41421i −1.00000 1.00000i 0.707107 + 0.707107i −2.82843 1.00000i 1.00000 + 2.00000i
17.2 0.707107 0.707107i −1.70711 + 0.292893i 1.00000i −0.707107 + 2.12132i −1.00000 + 1.41421i −1.00000 1.00000i −0.707107 0.707107i 2.82843 1.00000i 1.00000 + 2.00000i
23.1 −0.707107 0.707107i −0.292893 1.70711i 1.00000i 0.707107 + 2.12132i −1.00000 + 1.41421i −1.00000 + 1.00000i 0.707107 0.707107i −2.82843 + 1.00000i 1.00000 2.00000i
23.2 0.707107 + 0.707107i −1.70711 0.292893i 1.00000i −0.707107 2.12132i −1.00000 1.41421i −1.00000 + 1.00000i −0.707107 + 0.707107i 2.82843 + 1.00000i 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
3.b Odd 1 yes
5.c Odd 1 yes
15.e Even 1 yes

Hecke kernels

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