Properties

Label 1815.2.a.x
Level $1815$
Weight $2$
Character orbit 1815.a
Self dual yes
Analytic conductor $14.493$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4928479669\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
Defining polynomial: \(x^{4} - x^{3} - 3 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 - \beta_{1} - \beta_{2} ) q^{2} + q^{3} + ( 3 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{4} - q^{5} + ( 2 - \beta_{1} - \beta_{2} ) q^{6} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{7} + ( 4 - \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{8} + q^{9} +O(q^{10})\) \( q + ( 2 - \beta_{1} - \beta_{2} ) q^{2} + q^{3} + ( 3 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{4} - q^{5} + ( 2 - \beta_{1} - \beta_{2} ) q^{6} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{7} + ( 4 - \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{8} + q^{9} + ( -2 + \beta_{1} + \beta_{2} ) q^{10} + ( 3 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{12} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{13} + ( 1 + 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{14} - q^{15} + ( 5 - 5 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} ) q^{16} + 5 q^{17} + ( 2 - \beta_{1} - \beta_{2} ) q^{18} + ( -1 - 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{19} + ( -3 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{20} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{21} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{23} + ( 4 - \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{24} + q^{25} + ( 1 + 2 \beta_{2} - \beta_{3} ) q^{26} + q^{27} + ( 4 + 5 \beta_{1} - 9 \beta_{3} ) q^{28} + ( -1 + \beta_{1} + 4 \beta_{2} ) q^{29} + ( -2 + \beta_{1} + \beta_{2} ) q^{30} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{31} + ( 7 - 8 \beta_{1} - 4 \beta_{2} + 9 \beta_{3} ) q^{32} + ( 10 - 5 \beta_{1} - 5 \beta_{2} ) q^{34} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{35} + ( 3 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{36} + ( \beta_{1} + \beta_{2} - 5 \beta_{3} ) q^{37} + ( -4 - 7 \beta_{1} + 3 \beta_{2} + 9 \beta_{3} ) q^{38} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{39} + ( -4 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{40} + ( 6 + 4 \beta_{1} + \beta_{2} - 5 \beta_{3} ) q^{41} + ( 1 + 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{42} + ( -3 - \beta_{2} + 6 \beta_{3} ) q^{43} - q^{45} + ( 1 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{46} + ( -6 - 4 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{47} + ( 5 - 5 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} ) q^{48} + ( 4 - 2 \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{49} + ( 2 - \beta_{1} - \beta_{2} ) q^{50} + 5 q^{51} + ( -2 + \beta_{1} ) q^{52} + ( 1 + 6 \beta_{1} + 3 \beta_{2} - 5 \beta_{3} ) q^{53} + ( 2 - \beta_{1} - \beta_{2} ) q^{54} + ( 6 + 8 \beta_{1} + 4 \beta_{2} - 15 \beta_{3} ) q^{56} + ( -1 - 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{57} + ( -6 - 3 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{58} + ( -5 + \beta_{1} + \beta_{2} + 6 \beta_{3} ) q^{59} + ( -3 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{60} + ( 3 - 3 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{61} + ( 4 \beta_{1} - \beta_{2} - 7 \beta_{3} ) q^{62} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{63} + ( 8 - 11 \beta_{1} - 2 \beta_{2} + 16 \beta_{3} ) q^{64} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{65} + ( -5 - 5 \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{67} + ( 15 - 5 \beta_{1} - 10 \beta_{2} + 5 \beta_{3} ) q^{68} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{69} + ( -1 - 3 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{70} + ( -4 + 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{71} + ( 4 - \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{72} + ( 2 - 3 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{73} + ( -1 + 9 \beta_{1} + 5 \beta_{2} - 11 \beta_{3} ) q^{74} + q^{75} + ( -9 - 11 \beta_{1} + \beta_{2} + 19 \beta_{3} ) q^{76} + ( 1 + 2 \beta_{2} - \beta_{3} ) q^{78} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - 7 \beta_{3} ) q^{79} + ( -5 + 5 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} ) q^{80} + q^{81} + ( 11 + 3 \beta_{1} - 4 \beta_{2} - 14 \beta_{3} ) q^{82} + ( 10 + 5 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} ) q^{83} + ( 4 + 5 \beta_{1} - 9 \beta_{3} ) q^{84} -5 q^{85} + ( -5 - 8 \beta_{1} - 4 \beta_{2} + 12 \beta_{3} ) q^{86} + ( -1 + \beta_{1} + 4 \beta_{2} ) q^{87} + ( 2 + 2 \beta_{1} + 3 \beta_{3} ) q^{89} + ( -2 + \beta_{1} + \beta_{2} ) q^{90} + ( -6 + 2 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} ) q^{91} + ( 1 + 2 \beta_{1} - 3 \beta_{3} ) q^{92} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{93} + ( -13 - \beta_{1} + 8 \beta_{2} + 10 \beta_{3} ) q^{94} + ( 1 + 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{95} + ( 7 - 8 \beta_{1} - 4 \beta_{2} + 9 \beta_{3} ) q^{96} + ( -1 + 4 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{97} + ( 7 + 3 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 5q^{2} + 4q^{3} + 9q^{4} - 4q^{5} + 5q^{6} - 2q^{7} + 15q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 5q^{2} + 4q^{3} + 9q^{4} - 4q^{5} + 5q^{6} - 2q^{7} + 15q^{8} + 4q^{9} - 5q^{10} + 9q^{12} + 3q^{13} - 5q^{14} - 4q^{15} + 15q^{16} + 20q^{17} + 5q^{18} + 3q^{19} - 9q^{20} - 2q^{21} - 5q^{23} + 15q^{24} + 4q^{25} + 6q^{26} + 4q^{27} + 3q^{28} + 5q^{29} - 5q^{30} - q^{31} + 30q^{32} + 25q^{34} + 2q^{35} + 9q^{36} - 7q^{37} + q^{38} + 3q^{39} - 15q^{40} + 20q^{41} - 5q^{42} - 2q^{43} - 4q^{45} + 7q^{46} - 20q^{47} + 15q^{48} + 8q^{49} + 5q^{50} + 20q^{51} - 7q^{52} + 6q^{53} + 5q^{54} + 10q^{56} + 3q^{57} - 21q^{58} - 5q^{59} - 9q^{60} - 7q^{61} - 12q^{62} - 2q^{63} + 49q^{64} - 3q^{65} - 13q^{67} + 45q^{68} - 5q^{69} + 5q^{70} - 25q^{71} + 15q^{72} + 23q^{73} - 7q^{74} + 4q^{75} - 7q^{76} + 6q^{78} - 15q^{80} + 4q^{81} + 11q^{82} + 33q^{83} + 3q^{84} - 20q^{85} - 12q^{86} + 5q^{87} + 16q^{89} - 5q^{90} - 24q^{91} - q^{93} - 17q^{94} - 3q^{95} + 30q^{96} + 25q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 3 x^{2} + x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 2 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 3 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.09529
−1.35567
0.737640
−0.477260
−1.39026 1.00000 −0.0671858 −1.00000 −1.39026 1.27759 2.87392 1.00000 1.39026
1.2 1.16215 1.00000 −0.649414 −1.00000 1.16215 −4.28684 −3.07901 1.00000 −1.16215
1.3 2.45589 1.00000 4.03138 −1.00000 2.45589 3.28684 4.98884 1.00000 −2.45589
1.4 2.77222 1.00000 5.68522 −1.00000 2.77222 −2.27759 10.2163 1.00000 −2.77222
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1815.2.a.x 4
3.b odd 2 1 5445.2.a.be 4
5.b even 2 1 9075.2.a.cl 4
11.b odd 2 1 1815.2.a.o 4
11.c even 5 2 165.2.m.a 8
33.d even 2 1 5445.2.a.bv 4
33.h odd 10 2 495.2.n.d 8
55.d odd 2 1 9075.2.a.dj 4
55.j even 10 2 825.2.n.k 8
55.k odd 20 4 825.2.bx.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.a 8 11.c even 5 2
495.2.n.d 8 33.h odd 10 2
825.2.n.k 8 55.j even 10 2
825.2.bx.h 16 55.k odd 20 4
1815.2.a.o 4 11.b odd 2 1
1815.2.a.x 4 1.a even 1 1 trivial
5445.2.a.be 4 3.b odd 2 1
5445.2.a.bv 4 33.d even 2 1
9075.2.a.cl 4 5.b even 2 1
9075.2.a.dj 4 55.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1815))\):

\( T_{2}^{4} - 5 T_{2}^{3} + 4 T_{2}^{2} + 10 T_{2} - 11 \)
\( T_{7}^{4} + 2 T_{7}^{3} - 16 T_{7}^{2} - 17 T_{7} + 41 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -11 + 10 T + 4 T^{2} - 5 T^{3} + T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( ( 1 + T )^{4} \)
$7$ \( 41 - 17 T - 16 T^{2} + 2 T^{3} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( -1 - 6 T - 10 T^{2} - 3 T^{3} + T^{4} \)
$17$ \( ( -5 + T )^{4} \)
$19$ \( -31 + 204 T - 50 T^{2} - 3 T^{3} + T^{4} \)
$23$ \( -1 - 5 T - T^{2} + 5 T^{3} + T^{4} \)
$29$ \( 539 + 140 T - 44 T^{2} - 5 T^{3} + T^{4} \)
$31$ \( 139 - 7 T - 25 T^{2} + T^{3} + T^{4} \)
$37$ \( 431 - 133 T - 37 T^{2} + 7 T^{3} + T^{4} \)
$41$ \( -2071 + 335 T + 86 T^{2} - 20 T^{3} + T^{4} \)
$43$ \( 1861 - 63 T - 92 T^{2} + 2 T^{3} + T^{4} \)
$47$ \( -11 - 25 T + 94 T^{2} + 20 T^{3} + T^{4} \)
$53$ \( -1271 + 777 T - 100 T^{2} - 6 T^{3} + T^{4} \)
$59$ \( 2299 - 325 T - 101 T^{2} + 5 T^{3} + T^{4} \)
$61$ \( 1891 - 322 T - 136 T^{2} + 7 T^{3} + T^{4} \)
$67$ \( -3379 - 1768 T - 136 T^{2} + 13 T^{3} + T^{4} \)
$71$ \( -2351 - 25 T + 171 T^{2} + 25 T^{3} + T^{4} \)
$73$ \( -9199 + 1722 T + 48 T^{2} - 23 T^{3} + T^{4} \)
$79$ \( 1199 + 210 T - 109 T^{2} + T^{4} \)
$83$ \( -12221 + 869 T + 265 T^{2} - 33 T^{3} + T^{4} \)
$89$ \( -271 + 132 T + 45 T^{2} - 16 T^{3} + T^{4} \)
$97$ \( -25 + 125 T - 60 T^{2} + T^{4} \)
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