Properties

Label 1815.4.a.l
Level $1815$
Weight $4$
Character orbit 1815.a
Self dual yes
Analytic conductor $107.088$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1815.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(107.088466660\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta q^{2} + 3 q^{3} + 4 q^{4} + 5 q^{5} + 6 \beta q^{6} + 18 \beta q^{7} -8 \beta q^{8} + 9 q^{9} +O(q^{10})\) \( q + 2 \beta q^{2} + 3 q^{3} + 4 q^{4} + 5 q^{5} + 6 \beta q^{6} + 18 \beta q^{7} -8 \beta q^{8} + 9 q^{9} + 10 \beta q^{10} + 12 q^{12} -44 \beta q^{13} + 108 q^{14} + 15 q^{15} -80 q^{16} + 27 \beta q^{17} + 18 \beta q^{18} -18 \beta q^{19} + 20 q^{20} + 54 \beta q^{21} + 195 q^{23} -24 \beta q^{24} + 25 q^{25} -264 q^{26} + 27 q^{27} + 72 \beta q^{28} + 20 \beta q^{29} + 30 \beta q^{30} + 97 q^{31} -96 \beta q^{32} + 162 q^{34} + 90 \beta q^{35} + 36 q^{36} + 274 q^{37} -108 q^{38} -132 \beta q^{39} -40 \beta q^{40} + 222 \beta q^{41} + 324 q^{42} + 240 \beta q^{43} + 45 q^{45} + 390 \beta q^{46} + 3 q^{47} -240 q^{48} + 629 q^{49} + 50 \beta q^{50} + 81 \beta q^{51} -176 \beta q^{52} + 3 q^{53} + 54 \beta q^{54} -432 q^{56} -54 \beta q^{57} + 120 q^{58} -648 q^{59} + 60 q^{60} -265 \beta q^{61} + 194 \beta q^{62} + 162 \beta q^{63} + 64 q^{64} -220 \beta q^{65} + 70 q^{67} + 108 \beta q^{68} + 585 q^{69} + 540 q^{70} -372 q^{71} -72 \beta q^{72} + 540 \beta q^{73} + 548 \beta q^{74} + 75 q^{75} -72 \beta q^{76} -792 q^{78} + 301 \beta q^{79} -400 q^{80} + 81 q^{81} + 1332 q^{82} + 26 \beta q^{83} + 216 \beta q^{84} + 135 \beta q^{85} + 1440 q^{86} + 60 \beta q^{87} + 1122 q^{89} + 90 \beta q^{90} -2376 q^{91} + 780 q^{92} + 291 q^{93} + 6 \beta q^{94} -90 \beta q^{95} -288 \beta q^{96} + 470 q^{97} + 1258 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{3} + 8q^{4} + 10q^{5} + 18q^{9} + O(q^{10}) \) \( 2q + 6q^{3} + 8q^{4} + 10q^{5} + 18q^{9} + 24q^{12} + 216q^{14} + 30q^{15} - 160q^{16} + 40q^{20} + 390q^{23} + 50q^{25} - 528q^{26} + 54q^{27} + 194q^{31} + 324q^{34} + 72q^{36} + 548q^{37} - 216q^{38} + 648q^{42} + 90q^{45} + 6q^{47} - 480q^{48} + 1258q^{49} + 6q^{53} - 864q^{56} + 240q^{58} - 1296q^{59} + 120q^{60} + 128q^{64} + 140q^{67} + 1170q^{69} + 1080q^{70} - 744q^{71} + 150q^{75} - 1584q^{78} - 800q^{80} + 162q^{81} + 2664q^{82} + 2880q^{86} + 2244q^{89} - 4752q^{91} + 1560q^{92} + 582q^{93} + 940q^{97} + O(q^{100}) \)

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
−1.73205
1.73205
−3.46410 3.00000 4.00000 5.00000 −10.3923 −31.1769 13.8564 9.00000 −17.3205
1.2 3.46410 3.00000 4.00000 5.00000 10.3923 31.1769 −13.8564 9.00000 17.3205
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1815.4.a.l 2
11.b odd 2 1 inner 1815.4.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.4.a.l 2 1.a even 1 1 trivial
1815.4.a.l 2 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1815))\):

\( T_{2}^{2} - 12 \)
\( T_{7}^{2} - 972 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -12 + T^{2} \)
$3$ \( ( -3 + T )^{2} \)
$5$ \( ( -5 + T )^{2} \)
$7$ \( -972 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( -5808 + T^{2} \)
$17$ \( -2187 + T^{2} \)
$19$ \( -972 + T^{2} \)
$23$ \( ( -195 + T )^{2} \)
$29$ \( -1200 + T^{2} \)
$31$ \( ( -97 + T )^{2} \)
$37$ \( ( -274 + T )^{2} \)
$41$ \( -147852 + T^{2} \)
$43$ \( -172800 + T^{2} \)
$47$ \( ( -3 + T )^{2} \)
$53$ \( ( -3 + T )^{2} \)
$59$ \( ( 648 + T )^{2} \)
$61$ \( -210675 + T^{2} \)
$67$ \( ( -70 + T )^{2} \)
$71$ \( ( 372 + T )^{2} \)
$73$ \( -874800 + T^{2} \)
$79$ \( -271803 + T^{2} \)
$83$ \( -2028 + T^{2} \)
$89$ \( ( -1122 + T )^{2} \)
$97$ \( ( -470 + T )^{2} \)
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