Properties

Label 30.6.c.a
Level $30$
Weight $6$
Character orbit 30.c
Analytic conductor $4.812$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 30.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.81151459439\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + 4 i q^{2} + 9 i q^{3} -16 q^{4} + ( -55 + 10 i ) q^{5} -36 q^{6} -4 i q^{7} -64 i q^{8} -81 q^{9} +O(q^{10})\) \( q + 4 i q^{2} + 9 i q^{3} -16 q^{4} + ( -55 + 10 i ) q^{5} -36 q^{6} -4 i q^{7} -64 i q^{8} -81 q^{9} + ( -40 - 220 i ) q^{10} -500 q^{11} -144 i q^{12} + 288 i q^{13} + 16 q^{14} + ( -90 - 495 i ) q^{15} + 256 q^{16} + 1516 i q^{17} -324 i q^{18} + 1344 q^{19} + ( 880 - 160 i ) q^{20} + 36 q^{21} -2000 i q^{22} + 4100 i q^{23} + 576 q^{24} + ( 2925 - 1100 i ) q^{25} -1152 q^{26} -729 i q^{27} + 64 i q^{28} + 2646 q^{29} + ( 1980 - 360 i ) q^{30} -5612 q^{31} + 1024 i q^{32} -4500 i q^{33} -6064 q^{34} + ( 40 + 220 i ) q^{35} + 1296 q^{36} -7288 i q^{37} + 5376 i q^{38} -2592 q^{39} + ( 640 + 3520 i ) q^{40} -18986 q^{41} + 144 i q^{42} + 2404 i q^{43} + 8000 q^{44} + ( 4455 - 810 i ) q^{45} -16400 q^{46} + 8900 i q^{47} + 2304 i q^{48} + 16791 q^{49} + ( 4400 + 11700 i ) q^{50} -13644 q^{51} -4608 i q^{52} -39804 i q^{53} + 2916 q^{54} + ( 27500 - 5000 i ) q^{55} -256 q^{56} + 12096 i q^{57} + 10584 i q^{58} + 28300 q^{59} + ( 1440 + 7920 i ) q^{60} + 18290 q^{61} -22448 i q^{62} + 324 i q^{63} -4096 q^{64} + ( -2880 - 15840 i ) q^{65} + 18000 q^{66} + 65956 i q^{67} -24256 i q^{68} -36900 q^{69} + ( -880 + 160 i ) q^{70} -28800 q^{71} + 5184 i q^{72} + 30808 i q^{73} + 29152 q^{74} + ( 9900 + 26325 i ) q^{75} -21504 q^{76} + 2000 i q^{77} -10368 i q^{78} -60228 q^{79} + ( -14080 + 2560 i ) q^{80} + 6561 q^{81} -75944 i q^{82} + 2468 i q^{83} -576 q^{84} + ( -15160 - 83380 i ) q^{85} -9616 q^{86} + 23814 i q^{87} + 32000 i q^{88} -22678 q^{89} + ( 3240 + 17820 i ) q^{90} + 1152 q^{91} -65600 i q^{92} -50508 i q^{93} -35600 q^{94} + ( -73920 + 13440 i ) q^{95} -9216 q^{96} -36968 i q^{97} + 67164 i q^{98} + 40500 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{4} - 110q^{5} - 72q^{6} - 162q^{9} + O(q^{10}) \) \( 2q - 32q^{4} - 110q^{5} - 72q^{6} - 162q^{9} - 80q^{10} - 1000q^{11} + 32q^{14} - 180q^{15} + 512q^{16} + 2688q^{19} + 1760q^{20} + 72q^{21} + 1152q^{24} + 5850q^{25} - 2304q^{26} + 5292q^{29} + 3960q^{30} - 11224q^{31} - 12128q^{34} + 80q^{35} + 2592q^{36} - 5184q^{39} + 1280q^{40} - 37972q^{41} + 16000q^{44} + 8910q^{45} - 32800q^{46} + 33582q^{49} + 8800q^{50} - 27288q^{51} + 5832q^{54} + 55000q^{55} - 512q^{56} + 56600q^{59} + 2880q^{60} + 36580q^{61} - 8192q^{64} - 5760q^{65} + 36000q^{66} - 73800q^{69} - 1760q^{70} - 57600q^{71} + 58304q^{74} + 19800q^{75} - 43008q^{76} - 120456q^{79} - 28160q^{80} + 13122q^{81} - 1152q^{84} - 30320q^{85} - 19232q^{86} - 45356q^{89} + 6480q^{90} + 2304q^{91} - 71200q^{94} - 147840q^{95} - 18432q^{96} + 81000q^{99} + O(q^{100}) \)

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
4.00000i 9.00000i −16.0000 −55.0000 10.0000i −36.0000 4.00000i 64.0000i −81.0000 −40.0000 + 220.000i
19.2 4.00000i 9.00000i −16.0000 −55.0000 + 10.0000i −36.0000 4.00000i 64.0000i −81.0000 −40.0000 220.000i
\(n\): 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 30.6.c.a 2
3.b odd 2 1 90.6.c.b 2
4.b odd 2 1 240.6.f.a 2
5.b even 2 1 inner 30.6.c.a 2
5.c odd 4 1 150.6.a.f 1
5.c odd 4 1 150.6.a.j 1
12.b even 2 1 720.6.f.g 2
15.d odd 2 1 90.6.c.b 2
15.e even 4 1 450.6.a.f 1
15.e even 4 1 450.6.a.s 1
20.d odd 2 1 240.6.f.a 2
60.h even 2 1 720.6.f.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.c.a 2 1.a even 1 1 trivial
30.6.c.a 2 5.b even 2 1 inner
90.6.c.b 2 3.b odd 2 1
90.6.c.b 2 15.d odd 2 1
150.6.a.f 1 5.c odd 4 1
150.6.a.j 1 5.c odd 4 1
240.6.f.a 2 4.b odd 2 1
240.6.f.a 2 20.d odd 2 1
450.6.a.f 1 15.e even 4 1
450.6.a.s 1 15.e even 4 1
720.6.f.g 2 12.b even 2 1
720.6.f.g 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 16 \) acting on \(S_{6}^{\mathrm{new}}(30, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + T^{2} \)
$3$ \( 81 + T^{2} \)
$5$ \( 3125 + 110 T + T^{2} \)
$7$ \( 16 + T^{2} \)
$11$ \( ( 500 + T )^{2} \)
$13$ \( 82944 + T^{2} \)
$17$ \( 2298256 + T^{2} \)
$19$ \( ( -1344 + T )^{2} \)
$23$ \( 16810000 + T^{2} \)
$29$ \( ( -2646 + T )^{2} \)
$31$ \( ( 5612 + T )^{2} \)
$37$ \( 53114944 + T^{2} \)
$41$ \( ( 18986 + T )^{2} \)
$43$ \( 5779216 + T^{2} \)
$47$ \( 79210000 + T^{2} \)
$53$ \( 1584358416 + T^{2} \)
$59$ \( ( -28300 + T )^{2} \)
$61$ \( ( -18290 + T )^{2} \)
$67$ \( 4350193936 + T^{2} \)
$71$ \( ( 28800 + T )^{2} \)
$73$ \( 949132864 + T^{2} \)
$79$ \( ( 60228 + T )^{2} \)
$83$ \( 6091024 + T^{2} \)
$89$ \( ( 22678 + T )^{2} \)
$97$ \( 1366633024 + T^{2} \)
show more
show less