Properties

Label 1815.2.a.p
Level $1815$
Weight $2$
Character orbit 1815.a
Self dual yes
Analytic conductor $14.493$
Analytic rank $1$
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: \(1\)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} + q^{3} + ( -\beta_{1} + \beta_{2} ) q^{4} + q^{5} + ( -1 + \beta_{1} ) q^{6} + ( -2 + \beta_{2} ) q^{7} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{8} + q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} + q^{3} + ( -\beta_{1} + \beta_{2} ) q^{4} + q^{5} + ( -1 + \beta_{1} ) q^{6} + ( -2 + \beta_{2} ) q^{7} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{8} + q^{9} + ( -1 + \beta_{1} ) q^{10} + ( -\beta_{1} + \beta_{2} ) q^{12} + ( -1 - \beta_{1} - \beta_{3} ) q^{13} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{14} + q^{15} + ( 1 + \beta_{1} - 3 \beta_{3} ) q^{16} + ( -1 - 2 \beta_{1} - 2 \beta_{2} ) q^{17} + ( -1 + \beta_{1} ) q^{18} + ( -2 + \beta_{1} - \beta_{2} ) q^{19} + ( -\beta_{1} + \beta_{2} ) q^{20} + ( -2 + \beta_{2} ) q^{21} + ( 2 - \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{23} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{24} + q^{25} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{26} + q^{27} + ( 3 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{28} + ( -3 + 5 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{29} + ( -1 + \beta_{1} ) q^{30} + ( -4 - 5 \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{31} + ( 2 \beta_{2} + \beta_{3} ) q^{32} + ( 1 - 3 \beta_{1} - 2 \beta_{3} ) q^{34} + ( -2 + \beta_{2} ) q^{35} + ( -\beta_{1} + \beta_{2} ) q^{36} + ( -4 + 3 \beta_{1} + 6 \beta_{2} - \beta_{3} ) q^{37} + ( 4 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{38} + ( -1 - \beta_{1} - \beta_{3} ) q^{39} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{40} + ( -8 + \beta_{2} + 4 \beta_{3} ) q^{41} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{42} + ( -4 + 5 \beta_{2} + 3 \beta_{3} ) q^{43} + q^{45} + ( 1 - 3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{46} + ( -2 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{47} + ( 1 + \beta_{1} - 3 \beta_{3} ) q^{48} + ( -1 - 3 \beta_{2} - \beta_{3} ) q^{49} + ( -1 + \beta_{1} ) q^{50} + ( -1 - 2 \beta_{1} - 2 \beta_{2} ) q^{51} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{52} + ( 1 - 3 \beta_{2} + 6 \beta_{3} ) q^{53} + ( -1 + \beta_{1} ) q^{54} + ( -3 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{56} + ( -2 + \beta_{1} - \beta_{2} ) q^{57} + ( 9 - 7 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{58} + ( -4 + 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{59} + ( -\beta_{1} + \beta_{2} ) q^{60} + ( -3 - 7 \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{61} + ( \beta_{2} - 6 \beta_{3} ) q^{62} + ( -2 + \beta_{2} ) q^{63} + ( -4 + \beta_{1} - \beta_{2} + 7 \beta_{3} ) q^{64} + ( -1 - \beta_{1} - \beta_{3} ) q^{65} + ( 3 + 3 \beta_{1} - 7 \beta_{3} ) q^{67} + ( -2 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{68} + ( 2 - \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{69} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{70} + ( -4 + 3 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} ) q^{71} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{72} + ( -1 - 5 \beta_{1} + 4 \beta_{3} ) q^{73} + ( 1 + \beta_{1} - 4 \beta_{2} + 7 \beta_{3} ) q^{74} + q^{75} + ( -5 + 3 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{76} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{78} + ( 1 + 2 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} ) q^{79} + ( 1 + \beta_{1} - 3 \beta_{3} ) q^{80} + q^{81} + ( 7 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{82} + ( \beta_{1} + 3 \beta_{2} - 5 \beta_{3} ) q^{83} + ( 3 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{84} + ( -1 - 2 \beta_{1} - 2 \beta_{2} ) q^{85} + ( -1 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{86} + ( -3 + 5 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{87} + ( -5 - 4 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{89} + ( -1 + \beta_{1} ) q^{90} + ( 3 - \beta_{2} + \beta_{3} ) q^{91} + ( -10 + 2 \beta_{1} + 7 \beta_{3} ) q^{92} + ( -4 - 5 \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{93} + ( 7 - \beta_{1} + 7 \beta_{2} - 3 \beta_{3} ) q^{94} + ( -2 + \beta_{1} - \beta_{2} ) q^{95} + ( 2 \beta_{2} + \beta_{3} ) q^{96} + ( -7 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{97} + ( 4 - 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 3q^{2} + 4q^{3} + q^{4} + 4q^{5} - 3q^{6} - 6q^{7} - 3q^{8} + 4q^{9} + O(q^{10}) \) \( 4q - 3q^{2} + 4q^{3} + q^{4} + 4q^{5} - 3q^{6} - 6q^{7} - 3q^{8} + 4q^{9} - 3q^{10} + q^{12} - 7q^{13} + 3q^{14} + 4q^{15} - q^{16} - 10q^{17} - 3q^{18} - 9q^{19} + q^{20} - 6q^{21} - 3q^{23} - 3q^{24} + 4q^{25} - 4q^{26} + 4q^{27} + 7q^{28} - 15q^{29} - 3q^{30} - 13q^{31} + 6q^{32} - 3q^{34} - 6q^{35} + q^{36} - 3q^{37} + 15q^{38} - 7q^{39} - 3q^{40} - 22q^{41} + 3q^{42} + 4q^{45} - q^{46} - 2q^{47} - q^{48} - 12q^{49} - 3q^{50} - 10q^{51} + 9q^{52} + 10q^{53} - 3q^{54} - 8q^{56} - 9q^{57} + 39q^{58} - 21q^{59} + q^{60} - 11q^{61} - 10q^{62} - 6q^{63} - 3q^{64} - 7q^{65} + q^{67} - 3q^{68} - 3q^{69} + 3q^{70} - 13q^{71} - 3q^{72} - q^{73} + 11q^{74} + 4q^{75} - 19q^{76} - 4q^{78} + 4q^{79} - q^{80} + 4q^{81} + 25q^{82} - 3q^{83} + 7q^{84} - 10q^{85} - 15q^{87} - 10q^{89} - 3q^{90} + 12q^{91} - 24q^{92} - 13q^{93} + 35q^{94} - 9q^{95} + 6q^{96} - 22q^{97} + 11q^{98} + O(q^{100}) \)

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
−1.35567
−0.477260
0.737640
2.09529
−2.35567 1.00000 3.54920 1.00000 −2.35567 0.193527 −3.64941 1.00000 −2.35567
1.2 −1.47726 1.00000 0.182297 1.00000 −1.47726 −2.29496 2.68522 1.00000 −1.47726
1.3 −0.262360 1.00000 −1.93117 1.00000 −0.262360 −3.19353 1.03138 1.00000 −0.262360
1.4 1.09529 1.00000 −0.800331 1.00000 1.09529 −0.705037 −3.06719 1.00000 1.09529
\(n\): 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.p 4
3.b odd 2 1 5445.2.a.bt 4
5.b even 2 1 9075.2.a.di 4
11.b odd 2 1 1815.2.a.w 4
11.c even 5 2 165.2.m.d 8
33.d even 2 1 5445.2.a.bf 4
33.h odd 10 2 495.2.n.a 8
55.d odd 2 1 9075.2.a.cm 4
55.j even 10 2 825.2.n.g 8
55.k odd 20 4 825.2.bx.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.d 8 11.c even 5 2
495.2.n.a 8 33.h odd 10 2
825.2.n.g 8 55.j even 10 2
825.2.bx.f 16 55.k odd 20 4
1815.2.a.p 4 1.a even 1 1 trivial
1815.2.a.w 4 11.b odd 2 1
5445.2.a.bf 4 33.d even 2 1
5445.2.a.bt 4 3.b odd 2 1
9075.2.a.cm 4 55.d odd 2 1
9075.2.a.di 4 5.b even 2 1

Hecke kernels

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

\( T_{2}^{4} + 3 T_{2}^{3} - 4 T_{2} - 1 \)
\( T_{7}^{4} + 6 T_{7}^{3} + 10 T_{7}^{2} + 3 T_{7} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - 4 T + 3 T^{3} + T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( ( -1 + T )^{4} \)
$7$ \( -1 + 3 T + 10 T^{2} + 6 T^{3} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( -11 - 6 T + 10 T^{2} + 7 T^{3} + T^{4} \)
$17$ \( -1 - 10 T + 16 T^{2} + 10 T^{3} + T^{4} \)
$19$ \( 1 + 16 T + 22 T^{2} + 9 T^{3} + T^{4} \)
$23$ \( 449 - 129 T - 55 T^{2} + 3 T^{3} + T^{4} \)
$29$ \( 499 - 410 T + 4 T^{2} + 15 T^{3} + T^{4} \)
$31$ \( -1249 - 607 T - 17 T^{2} + 13 T^{3} + T^{4} \)
$37$ \( 3541 - 213 T - 121 T^{2} + 3 T^{3} + T^{4} \)
$41$ \( 41 + 207 T + 138 T^{2} + 22 T^{3} + T^{4} \)
$43$ \( 275 + 375 T - 110 T^{2} + T^{4} \)
$47$ \( 31 + 47 T - 92 T^{2} + 2 T^{3} + T^{4} \)
$53$ \( -641 + 815 T - 84 T^{2} - 10 T^{3} + T^{4} \)
$59$ \( -589 + 29 T + 117 T^{2} + 21 T^{3} + T^{4} \)
$61$ \( 1151 - 866 T - 88 T^{2} + 11 T^{3} + T^{4} \)
$67$ \( 619 + 112 T - 100 T^{2} - T^{3} + T^{4} \)
$71$ \( 281 - 387 T - 57 T^{2} + 13 T^{3} + T^{4} \)
$73$ \( 931 - 84 T - 74 T^{2} + T^{3} + T^{4} \)
$79$ \( -169 + 286 T - 89 T^{2} - 4 T^{3} + T^{4} \)
$83$ \( 41 - 147 T - 77 T^{2} + 3 T^{3} + T^{4} \)
$89$ \( 209 - 310 T - 89 T^{2} + 10 T^{3} + T^{4} \)
$97$ \( -1159 - 223 T + 88 T^{2} + 22 T^{3} + T^{4} \)
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