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: \( x^{4} - x^{3} - 3x^{2} + x + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1 + 2) q^{2} + q^{3} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 3) q^{4} - q^{5} + ( - \beta_{2} - \beta_1 + 2) q^{6} + ( - \beta_{3} - \beta_{2} + 2 \beta_1) q^{7} + (3 \beta_{3} - 3 \beta_{2} - \beta_1 + 4) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1 + 2) q^{2} + q^{3} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 3) q^{4} - q^{5} + ( - \beta_{2} - \beta_1 + 2) q^{6} + ( - \beta_{3} - \beta_{2} + 2 \beta_1) q^{7} + (3 \beta_{3} - 3 \beta_{2} - \beta_1 + 4) q^{8} + q^{9} + (\beta_{2} + \beta_1 - 2) q^{10} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 3) q^{12} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{13} + ( - 4 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 1) q^{14} - q^{15} + (5 \beta_{3} - 5 \beta_{2} - 5 \beta_1 + 5) q^{16} + 5 q^{17} + ( - \beta_{2} - \beta_1 + 2) q^{18} + (3 \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 1) q^{19} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{20} + ( - \beta_{3} - \beta_{2} + 2 \beta_1) q^{21} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{23} + (3 \beta_{3} - 3 \beta_{2} - \beta_1 + 4) q^{24} + q^{25} + ( - \beta_{3} + 2 \beta_{2} + 1) q^{26} + q^{27} + ( - 9 \beta_{3} + 5 \beta_1 + 4) q^{28} + (4 \beta_{2} + \beta_1 - 1) q^{29} + (\beta_{2} + \beta_1 - 2) q^{30} + ( - 2 \beta_{3} + 3 \beta_1) q^{31} + (9 \beta_{3} - 4 \beta_{2} - 8 \beta_1 + 7) q^{32} + ( - 5 \beta_{2} - 5 \beta_1 + 10) q^{34} + (\beta_{3} + \beta_{2} - 2 \beta_1) q^{35} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 3) q^{36} + ( - 5 \beta_{3} + \beta_{2} + \beta_1) q^{37} + (9 \beta_{3} + 3 \beta_{2} - 7 \beta_1 - 4) q^{38} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{39} + ( - 3 \beta_{3} + 3 \beta_{2} + \beta_1 - 4) q^{40} + ( - 5 \beta_{3} + \beta_{2} + 4 \beta_1 + 6) q^{41} + ( - 4 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 1) q^{42} + (6 \beta_{3} - \beta_{2} - 3) q^{43} - q^{45} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 + 1) q^{46} + (3 \beta_{3} + \beta_{2} - 4 \beta_1 - 6) q^{47} + (5 \beta_{3} - 5 \beta_{2} - 5 \beta_1 + 5) q^{48} + ( - 4 \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{49} + ( - \beta_{2} - \beta_1 + 2) q^{50} + 5 q^{51} + (\beta_1 - 2) q^{52} + ( - 5 \beta_{3} + 3 \beta_{2} + 6 \beta_1 + 1) q^{53} + ( - \beta_{2} - \beta_1 + 2) q^{54} + ( - 15 \beta_{3} + 4 \beta_{2} + 8 \beta_1 + 6) q^{56} + (3 \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 1) q^{57} + ( - \beta_{3} + 4 \beta_{2} - 3 \beta_1 - 6) q^{58} + (6 \beta_{3} + \beta_{2} + \beta_1 - 5) q^{59} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{60} + ( - 2 \beta_{3} - 6 \beta_{2} - 3 \beta_1 + 3) q^{61} + ( - 7 \beta_{3} - \beta_{2} + 4 \beta_1) q^{62} + ( - \beta_{3} - \beta_{2} + 2 \beta_1) q^{63} + (16 \beta_{3} - 2 \beta_{2} - 11 \beta_1 + 8) q^{64} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{65} + (\beta_{3} + 5 \beta_{2} - 5 \beta_1 - 5) q^{67} + (5 \beta_{3} - 10 \beta_{2} - 5 \beta_1 + 15) q^{68} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{69} + (4 \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 1) q^{70} + ( - 4 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 4) q^{71} + (3 \beta_{3} - 3 \beta_{2} - \beta_1 + 4) q^{72} + (4 \beta_{3} + 5 \beta_{2} - 3 \beta_1 + 2) q^{73} + ( - 11 \beta_{3} + 5 \beta_{2} + 9 \beta_1 - 1) q^{74} + q^{75} + (19 \beta_{3} + \beta_{2} - 11 \beta_1 - 9) q^{76} + ( - \beta_{3} + 2 \beta_{2} + 1) q^{78} + ( - 7 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{79} + ( - 5 \beta_{3} + 5 \beta_{2} + 5 \beta_1 - 5) q^{80} + q^{81} + ( - 14 \beta_{3} - 4 \beta_{2} + 3 \beta_1 + 11) q^{82} + ( - 8 \beta_{3} + 2 \beta_{2} + 5 \beta_1 + 10) q^{83} + ( - 9 \beta_{3} + 5 \beta_1 + 4) q^{84} - 5 q^{85} + (12 \beta_{3} - 4 \beta_{2} - 8 \beta_1 - 5) q^{86} + (4 \beta_{2} + \beta_1 - 1) q^{87} + (3 \beta_{3} + 2 \beta_1 + 2) q^{89} + (\beta_{2} + \beta_1 - 2) q^{90} + (4 \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 6) q^{91} + ( - 3 \beta_{3} + 2 \beta_1 + 1) q^{92} + ( - 2 \beta_{3} + 3 \beta_1) q^{93} + (10 \beta_{3} + 8 \beta_{2} - \beta_1 - 13) q^{94} + ( - 3 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 1) q^{95} + (9 \beta_{3} - 4 \beta_{2} - 8 \beta_1 + 7) q^{96} + ( - 3 \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 1) q^{97} + ( - 6 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 7) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{2} + 4 q^{3} + 9 q^{4} - 4 q^{5} + 5 q^{6} - 2 q^{7} + 15 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 5 q^{2} + 4 q^{3} + 9 q^{4} - 4 q^{5} + 5 q^{6} - 2 q^{7} + 15 q^{8} + 4 q^{9} - 5 q^{10} + 9 q^{12} + 3 q^{13} - 5 q^{14} - 4 q^{15} + 15 q^{16} + 20 q^{17} + 5 q^{18} + 3 q^{19} - 9 q^{20} - 2 q^{21} - 5 q^{23} + 15 q^{24} + 4 q^{25} + 6 q^{26} + 4 q^{27} + 3 q^{28} + 5 q^{29} - 5 q^{30} - q^{31} + 30 q^{32} + 25 q^{34} + 2 q^{35} + 9 q^{36} - 7 q^{37} + q^{38} + 3 q^{39} - 15 q^{40} + 20 q^{41} - 5 q^{42} - 2 q^{43} - 4 q^{45} + 7 q^{46} - 20 q^{47} + 15 q^{48} + 8 q^{49} + 5 q^{50} + 20 q^{51} - 7 q^{52} + 6 q^{53} + 5 q^{54} + 10 q^{56} + 3 q^{57} - 21 q^{58} - 5 q^{59} - 9 q^{60} - 7 q^{61} - 12 q^{62} - 2 q^{63} + 49 q^{64} - 3 q^{65} - 13 q^{67} + 45 q^{68} - 5 q^{69} + 5 q^{70} - 25 q^{71} + 15 q^{72} + 23 q^{73} - 7 q^{74} + 4 q^{75} - 7 q^{76} + 6 q^{78} - 15 q^{80} + 4 q^{81} + 11 q^{82} + 33 q^{83} + 3 q^{84} - 20 q^{85} - 12 q^{86} + 5 q^{87} + 16 q^{89} - 5 q^{90} - 24 q^{91} - q^{93} - 17 q^{94} - 3 q^{95} + 30 q^{96} + 25 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 2\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 3\beta_1 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1815))\):

\( T_{2}^{4} - 5T_{2}^{3} + 4T_{2}^{2} + 10T_{2} - 11 \) Copy content Toggle raw display
\( T_{7}^{4} + 2T_{7}^{3} - 16T_{7}^{2} - 17T_{7} + 41 \) Copy content Toggle raw display

Hecke characteristic polynomials

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