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 \(2\), degree \(1\), minimal)

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: \( 1 \)
Twist minimal: yes
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 Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 30.2.c.a 2
3.b odd 2 1 90.2.c.a 2
4.b odd 2 1 240.2.f.a 2
5.b even 2 1 inner 30.2.c.a 2
5.c odd 4 1 150.2.a.a 1
5.c odd 4 1 150.2.a.c 1
7.b odd 2 1 1470.2.g.g 2
7.c even 3 2 1470.2.n.h 4
7.d odd 6 2 1470.2.n.a 4
8.b even 2 1 960.2.f.h 2
8.d odd 2 1 960.2.f.i 2
9.c even 3 2 810.2.i.e 4
9.d odd 6 2 810.2.i.b 4
12.b even 2 1 720.2.f.f 2
15.d odd 2 1 90.2.c.a 2
15.e even 4 1 450.2.a.b 1
15.e even 4 1 450.2.a.f 1
16.e even 4 1 3840.2.d.g 2
16.e even 4 1 3840.2.d.y 2
16.f odd 4 1 3840.2.d.j 2
16.f odd 4 1 3840.2.d.x 2
20.d odd 2 1 240.2.f.a 2
20.e even 4 1 1200.2.a.g 1
20.e even 4 1 1200.2.a.m 1
24.f even 2 1 2880.2.f.c 2
24.h odd 2 1 2880.2.f.e 2
35.c odd 2 1 1470.2.g.g 2
35.f even 4 1 7350.2.a.bg 1
35.f even 4 1 7350.2.a.cc 1
35.i odd 6 2 1470.2.n.a 4
35.j even 6 2 1470.2.n.h 4
40.e odd 2 1 960.2.f.i 2
40.f even 2 1 960.2.f.h 2
40.i odd 4 1 4800.2.a.l 1
40.i odd 4 1 4800.2.a.cg 1
40.k even 4 1 4800.2.a.m 1
40.k even 4 1 4800.2.a.cj 1
45.h odd 6 2 810.2.i.b 4
45.j even 6 2 810.2.i.e 4
60.h even 2 1 720.2.f.f 2
60.l odd 4 1 3600.2.a.o 1
60.l odd 4 1 3600.2.a.bg 1
80.k odd 4 1 3840.2.d.j 2
80.k odd 4 1 3840.2.d.x 2
80.q even 4 1 3840.2.d.g 2
80.q even 4 1 3840.2.d.y 2
120.i odd 2 1 2880.2.f.e 2
120.m even 2 1 2880.2.f.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.c.a 2 1.a even 1 1 trivial
30.2.c.a 2 5.b even 2 1 inner
90.2.c.a 2 3.b odd 2 1
90.2.c.a 2 15.d odd 2 1
150.2.a.a 1 5.c odd 4 1
150.2.a.c 1 5.c odd 4 1
240.2.f.a 2 4.b odd 2 1
240.2.f.a 2 20.d odd 2 1
450.2.a.b 1 15.e even 4 1
450.2.a.f 1 15.e even 4 1
720.2.f.f 2 12.b even 2 1
720.2.f.f 2 60.h even 2 1
810.2.i.b 4 9.d odd 6 2
810.2.i.b 4 45.h odd 6 2
810.2.i.e 4 9.c even 3 2
810.2.i.e 4 45.j even 6 2
960.2.f.h 2 8.b even 2 1
960.2.f.h 2 40.f even 2 1
960.2.f.i 2 8.d odd 2 1
960.2.f.i 2 40.e odd 2 1
1200.2.a.g 1 20.e even 4 1
1200.2.a.m 1 20.e even 4 1
1470.2.g.g 2 7.b odd 2 1
1470.2.g.g 2 35.c odd 2 1
1470.2.n.a 4 7.d odd 6 2
1470.2.n.a 4 35.i odd 6 2
1470.2.n.h 4 7.c even 3 2
1470.2.n.h 4 35.j even 6 2
2880.2.f.c 2 24.f even 2 1
2880.2.f.c 2 120.m even 2 1
2880.2.f.e 2 24.h odd 2 1
2880.2.f.e 2 120.i odd 2 1
3600.2.a.o 1 60.l odd 4 1
3600.2.a.bg 1 60.l odd 4 1
3840.2.d.g 2 16.e even 4 1
3840.2.d.g 2 80.q even 4 1
3840.2.d.j 2 16.f odd 4 1
3840.2.d.j 2 80.k odd 4 1
3840.2.d.x 2 16.f odd 4 1
3840.2.d.x 2 80.k odd 4 1
3840.2.d.y 2 16.e even 4 1
3840.2.d.y 2 80.q even 4 1
4800.2.a.l 1 40.i odd 4 1
4800.2.a.m 1 40.k even 4 1
4800.2.a.cg 1 40.i odd 4 1
4800.2.a.cj 1 40.k even 4 1
7350.2.a.bg 1 35.f even 4 1
7350.2.a.cc 1 35.f even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(30, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( 1 + 4 T + 5 T^{2} \)
$7$ \( 1 - 10 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 2 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \)
$17$ \( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \)
$19$ \( ( 1 + 19 T^{2} )^{2} \)
$23$ \( 1 - 30 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 8 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \)
$41$ \( ( 1 - 2 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 30 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 70 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 10 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 2 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 70 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 130 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 + 79 T^{2} )^{2} \)
$83$ \( 1 - 150 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 10 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} ) \)
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