Properties

Label 1815.2.c.b
Level $1815$
Weight $2$
Character orbit 1815.c
Analytic conductor $14.493$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.4928479669\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
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.

\(f(q)\) \(=\) \( q + i q^{2} -i q^{3} + q^{4} + ( -1 + 2 i ) q^{5} + q^{6} + 4 i q^{7} + 3 i q^{8} - q^{9} +O(q^{10})\) \( q + i q^{2} -i q^{3} + q^{4} + ( -1 + 2 i ) q^{5} + q^{6} + 4 i q^{7} + 3 i q^{8} - q^{9} + ( -2 - i ) q^{10} -i q^{12} -2 i q^{13} -4 q^{14} + ( 2 + i ) q^{15} - q^{16} + 2 i q^{17} -i q^{18} + 8 q^{19} + ( -1 + 2 i ) q^{20} + 4 q^{21} + 4 i q^{23} + 3 q^{24} + ( -3 - 4 i ) q^{25} + 2 q^{26} + i q^{27} + 4 i q^{28} + 4 q^{29} + ( -1 + 2 i ) q^{30} -8 q^{31} + 5 i q^{32} -2 q^{34} + ( -8 - 4 i ) q^{35} - q^{36} -8 i q^{37} + 8 i q^{38} -2 q^{39} + ( -6 - 3 i ) q^{40} -12 q^{41} + 4 i q^{42} + 8 i q^{43} + ( 1 - 2 i ) q^{45} -4 q^{46} + 4 i q^{47} + i q^{48} -9 q^{49} + ( 4 - 3 i ) q^{50} + 2 q^{51} -2 i q^{52} -4 i q^{53} - q^{54} -12 q^{56} -8 i q^{57} + 4 i q^{58} + 8 q^{59} + ( 2 + i ) q^{60} -8 i q^{62} -4 i q^{63} -7 q^{64} + ( 4 + 2 i ) q^{65} + 4 i q^{67} + 2 i q^{68} + 4 q^{69} + ( 4 - 8 i ) q^{70} -12 q^{71} -3 i q^{72} -2 i q^{73} + 8 q^{74} + ( -4 + 3 i ) q^{75} + 8 q^{76} -2 i q^{78} + 8 q^{79} + ( 1 - 2 i ) q^{80} + q^{81} -12 i q^{82} -4 i q^{83} + 4 q^{84} + ( -4 - 2 i ) q^{85} -8 q^{86} -4 i q^{87} -6 q^{89} + ( 2 + i ) q^{90} + 8 q^{91} + 4 i q^{92} + 8 i q^{93} -4 q^{94} + ( -8 + 16 i ) q^{95} + 5 q^{96} + 8 i q^{97} -9 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} - 2q^{5} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{4} - 2q^{5} + 2q^{6} - 2q^{9} - 4q^{10} - 8q^{14} + 4q^{15} - 2q^{16} + 16q^{19} - 2q^{20} + 8q^{21} + 6q^{24} - 6q^{25} + 4q^{26} + 8q^{29} - 2q^{30} - 16q^{31} - 4q^{34} - 16q^{35} - 2q^{36} - 4q^{39} - 12q^{40} - 24q^{41} + 2q^{45} - 8q^{46} - 18q^{49} + 8q^{50} + 4q^{51} - 2q^{54} - 24q^{56} + 16q^{59} + 4q^{60} - 14q^{64} + 8q^{65} + 8q^{69} + 8q^{70} - 24q^{71} + 16q^{74} - 8q^{75} + 16q^{76} + 16q^{79} + 2q^{80} + 2q^{81} + 8q^{84} - 8q^{85} - 16q^{86} - 12q^{89} + 4q^{90} + 16q^{91} - 8q^{94} - 16q^{95} + 10q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1815\mathbb{Z}\right)^\times\).

\(n\) \(727\) \(1211\) \(1696\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
364.1
1.00000i
1.00000i
1.00000i 1.00000i 1.00000 −1.00000 2.00000i 1.00000 4.00000i 3.00000i −1.00000 −2.00000 + 1.00000i
364.2 1.00000i 1.00000i 1.00000 −1.00000 + 2.00000i 1.00000 4.00000i 3.00000i −1.00000 −2.00000 1.00000i
\(n\): 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 1815.2.c.b yes 2
5.b even 2 1 inner 1815.2.c.b yes 2
5.c odd 4 1 9075.2.a.c 1
5.c odd 4 1 9075.2.a.r 1
11.b odd 2 1 1815.2.c.a 2
55.d odd 2 1 1815.2.c.a 2
55.e even 4 1 9075.2.a.f 1
55.e even 4 1 9075.2.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.c.a 2 11.b odd 2 1
1815.2.c.a 2 55.d odd 2 1
1815.2.c.b yes 2 1.a even 1 1 trivial
1815.2.c.b yes 2 5.b even 2 1 inner
9075.2.a.c 1 5.c odd 4 1
9075.2.a.f 1 55.e even 4 1
9075.2.a.o 1 55.e even 4 1
9075.2.a.r 1 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1815, [\chi])\):

\( T_{2}^{2} + 1 \)
\( T_{19} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( 5 + 2 T + T^{2} \)
$7$ \( 16 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( -8 + T )^{2} \)
$23$ \( 16 + T^{2} \)
$29$ \( ( -4 + T )^{2} \)
$31$ \( ( 8 + T )^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( ( 12 + T )^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( 16 + T^{2} \)
$53$ \( 16 + T^{2} \)
$59$ \( ( -8 + T )^{2} \)
$61$ \( T^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( 4 + T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 16 + T^{2} \)
$89$ \( ( 6 + T )^{2} \)
$97$ \( 64 + T^{2} \)
show more
show less