Properties

Label 1815.4.a.m
Level $1815$
Weight $4$
Character orbit 1815.a
Self dual yes
Analytic conductor $107.088$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{15}) \)
Defining polynomial: \(x^{2} - 15\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \sqrt{15}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 3 q^{3} + 7 q^{4} -5 q^{5} + 3 \beta q^{6} -4 \beta q^{7} -\beta q^{8} + 9 q^{9} +O(q^{10})\) \( q + \beta q^{2} + 3 q^{3} + 7 q^{4} -5 q^{5} + 3 \beta q^{6} -4 \beta q^{7} -\beta q^{8} + 9 q^{9} -5 \beta q^{10} + 21 q^{12} + 18 \beta q^{13} -60 q^{14} -15 q^{15} -71 q^{16} -4 \beta q^{17} + 9 \beta q^{18} + 2 \beta q^{19} -35 q^{20} -12 \beta q^{21} + 48 q^{23} -3 \beta q^{24} + 25 q^{25} + 270 q^{26} + 27 q^{27} -28 \beta q^{28} -54 \beta q^{29} -15 \beta q^{30} + 160 q^{31} -63 \beta q^{32} -60 q^{34} + 20 \beta q^{35} + 63 q^{36} -266 q^{37} + 30 q^{38} + 54 \beta q^{39} + 5 \beta q^{40} + 46 \beta q^{41} -180 q^{42} -80 \beta q^{43} -45 q^{45} + 48 \beta q^{46} -504 q^{47} -213 q^{48} -103 q^{49} + 25 \beta q^{50} -12 \beta q^{51} + 126 \beta q^{52} -342 q^{53} + 27 \beta q^{54} + 60 q^{56} + 6 \beta q^{57} -810 q^{58} -660 q^{59} -105 q^{60} -56 \beta q^{61} + 160 \beta q^{62} -36 \beta q^{63} -377 q^{64} -90 \beta q^{65} + 496 q^{67} -28 \beta q^{68} + 144 q^{69} + 300 q^{70} -708 q^{71} -9 \beta q^{72} -166 \beta q^{73} -266 \beta q^{74} + 75 q^{75} + 14 \beta q^{76} + 810 q^{78} -46 \beta q^{79} + 355 q^{80} + 81 q^{81} + 690 q^{82} -22 \beta q^{83} -84 \beta q^{84} + 20 \beta q^{85} -1200 q^{86} -162 \beta q^{87} -606 q^{89} -45 \beta q^{90} -1080 q^{91} + 336 q^{92} + 480 q^{93} -504 \beta q^{94} -10 \beta q^{95} -189 \beta q^{96} + 254 q^{97} -103 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{3} + 14q^{4} - 10q^{5} + 18q^{9} + O(q^{10}) \) \( 2q + 6q^{3} + 14q^{4} - 10q^{5} + 18q^{9} + 42q^{12} - 120q^{14} - 30q^{15} - 142q^{16} - 70q^{20} + 96q^{23} + 50q^{25} + 540q^{26} + 54q^{27} + 320q^{31} - 120q^{34} + 126q^{36} - 532q^{37} + 60q^{38} - 360q^{42} - 90q^{45} - 1008q^{47} - 426q^{48} - 206q^{49} - 684q^{53} + 120q^{56} - 1620q^{58} - 1320q^{59} - 210q^{60} - 754q^{64} + 992q^{67} + 288q^{69} + 600q^{70} - 1416q^{71} + 150q^{75} + 1620q^{78} + 710q^{80} + 162q^{81} + 1380q^{82} - 2400q^{86} - 1212q^{89} - 2160q^{91} + 672q^{92} + 960q^{93} + 508q^{97} + O(q^{100}) \)

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
−3.87298
3.87298
−3.87298 3.00000 7.00000 −5.00000 −11.6190 15.4919 3.87298 9.00000 19.3649
1.2 3.87298 3.00000 7.00000 −5.00000 11.6190 −15.4919 −3.87298 9.00000 −19.3649
\(n\): 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.m 2
11.b odd 2 1 inner 1815.4.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.4.a.m 2 1.a even 1 1 trivial
1815.4.a.m 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1815))\):

\( T_{2}^{2} - 15 \)
\( T_{7}^{2} - 240 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -15 + T^{2} \)
$3$ \( ( -3 + T )^{2} \)
$5$ \( ( 5 + T )^{2} \)
$7$ \( -240 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( -4860 + T^{2} \)
$17$ \( -240 + T^{2} \)
$19$ \( -60 + T^{2} \)
$23$ \( ( -48 + T )^{2} \)
$29$ \( -43740 + T^{2} \)
$31$ \( ( -160 + T )^{2} \)
$37$ \( ( 266 + T )^{2} \)
$41$ \( -31740 + T^{2} \)
$43$ \( -96000 + T^{2} \)
$47$ \( ( 504 + T )^{2} \)
$53$ \( ( 342 + T )^{2} \)
$59$ \( ( 660 + T )^{2} \)
$61$ \( -47040 + T^{2} \)
$67$ \( ( -496 + T )^{2} \)
$71$ \( ( 708 + T )^{2} \)
$73$ \( -413340 + T^{2} \)
$79$ \( -31740 + T^{2} \)
$83$ \( -7260 + T^{2} \)
$89$ \( ( 606 + T )^{2} \)
$97$ \( ( -254 + T )^{2} \)
show more
show less