Properties

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

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: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 3 q^{3} + ( 2 + \beta ) q^{4} + 5 q^{5} + 3 \beta q^{6} + ( 2 - 3 \beta ) q^{7} + ( 10 - 5 \beta ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + \beta q^{2} + 3 q^{3} + ( 2 + \beta ) q^{4} + 5 q^{5} + 3 \beta q^{6} + ( 2 - 3 \beta ) q^{7} + ( 10 - 5 \beta ) q^{8} + 9 q^{9} + 5 \beta q^{10} + ( 6 + 3 \beta ) q^{12} + ( 24 - \beta ) q^{13} + ( -30 - \beta ) q^{14} + 15 q^{15} + ( -66 - 3 \beta ) q^{16} + ( -47 - 4 \beta ) q^{17} + 9 \beta q^{18} + ( -42 + 9 \beta ) q^{19} + ( 10 + 5 \beta ) q^{20} + ( 6 - 9 \beta ) q^{21} + ( -13 - 31 \beta ) q^{23} + ( 30 - 15 \beta ) q^{24} + 25 q^{25} + ( -10 + 23 \beta ) q^{26} + 27 q^{27} + ( -26 - 7 \beta ) q^{28} + ( -60 - 7 \beta ) q^{29} + 15 \beta q^{30} + ( -81 - 21 \beta ) q^{31} + ( -110 - 29 \beta ) q^{32} + ( -40 - 51 \beta ) q^{34} + ( 10 - 15 \beta ) q^{35} + ( 18 + 9 \beta ) q^{36} + ( -154 + 79 \beta ) q^{37} + ( 90 - 33 \beta ) q^{38} + ( 72 - 3 \beta ) q^{39} + ( 50 - 25 \beta ) q^{40} + ( 6 + 6 \beta ) q^{41} + ( -90 - 3 \beta ) q^{42} + ( 66 + 54 \beta ) q^{43} + 45 q^{45} + ( -310 - 44 \beta ) q^{46} + ( -275 - 65 \beta ) q^{47} + ( -198 - 9 \beta ) q^{48} + ( -249 - 3 \beta ) q^{49} + 25 \beta q^{50} + ( -141 - 12 \beta ) q^{51} + ( 38 + 21 \beta ) q^{52} + ( -463 - \beta ) q^{53} + 27 \beta q^{54} + ( 170 - 25 \beta ) q^{56} + ( -126 + 27 \beta ) q^{57} + ( -70 - 67 \beta ) q^{58} + ( -278 + 176 \beta ) q^{59} + ( 30 + 15 \beta ) q^{60} + ( 281 - 53 \beta ) q^{61} + ( -210 - 102 \beta ) q^{62} + ( 18 - 27 \beta ) q^{63} + ( 238 - 115 \beta ) q^{64} + ( 120 - 5 \beta ) q^{65} + ( -644 + 150 \beta ) q^{67} + ( -134 - 59 \beta ) q^{68} + ( -39 - 93 \beta ) q^{69} + ( -150 - 5 \beta ) q^{70} + ( -74 - 125 \beta ) q^{71} + ( 90 - 45 \beta ) q^{72} + ( 204 + 32 \beta ) q^{73} + ( 790 - 75 \beta ) q^{74} + 75 q^{75} + ( 6 - 15 \beta ) q^{76} + ( -30 + 69 \beta ) q^{78} + ( 521 - 69 \beta ) q^{79} + ( -330 - 15 \beta ) q^{80} + 81 q^{81} + ( 60 + 12 \beta ) q^{82} + ( 90 - 165 \beta ) q^{83} + ( -78 - 21 \beta ) q^{84} + ( -235 - 20 \beta ) q^{85} + ( 540 + 120 \beta ) q^{86} + ( -180 - 21 \beta ) q^{87} + ( -672 + 56 \beta ) q^{89} + 45 \beta q^{90} + ( 78 - 71 \beta ) q^{91} + ( -336 - 106 \beta ) q^{92} + ( -243 - 63 \beta ) q^{93} + ( -650 - 340 \beta ) q^{94} + ( -210 + 45 \beta ) q^{95} + ( -330 - 87 \beta ) q^{96} + ( -12 - 52 \beta ) q^{97} + ( -30 - 252 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 6q^{3} + 5q^{4} + 10q^{5} + 3q^{6} + q^{7} + 15q^{8} + 18q^{9} + O(q^{10}) \) \( 2q + q^{2} + 6q^{3} + 5q^{4} + 10q^{5} + 3q^{6} + q^{7} + 15q^{8} + 18q^{9} + 5q^{10} + 15q^{12} + 47q^{13} - 61q^{14} + 30q^{15} - 135q^{16} - 98q^{17} + 9q^{18} - 75q^{19} + 25q^{20} + 3q^{21} - 57q^{23} + 45q^{24} + 50q^{25} + 3q^{26} + 54q^{27} - 59q^{28} - 127q^{29} + 15q^{30} - 183q^{31} - 249q^{32} - 131q^{34} + 5q^{35} + 45q^{36} - 229q^{37} + 147q^{38} + 141q^{39} + 75q^{40} + 18q^{41} - 183q^{42} + 186q^{43} + 90q^{45} - 664q^{46} - 615q^{47} - 405q^{48} - 501q^{49} + 25q^{50} - 294q^{51} + 97q^{52} - 927q^{53} + 27q^{54} + 315q^{56} - 225q^{57} - 207q^{58} - 380q^{59} + 75q^{60} + 509q^{61} - 522q^{62} + 9q^{63} + 361q^{64} + 235q^{65} - 1138q^{67} - 327q^{68} - 171q^{69} - 305q^{70} - 273q^{71} + 135q^{72} + 440q^{73} + 1505q^{74} + 150q^{75} - 3q^{76} + 9q^{78} + 973q^{79} - 675q^{80} + 162q^{81} + 132q^{82} + 15q^{83} - 177q^{84} - 490q^{85} + 1200q^{86} - 381q^{87} - 1288q^{89} + 45q^{90} + 85q^{91} - 778q^{92} - 549q^{93} - 1640q^{94} - 375q^{95} - 747q^{96} - 76q^{97} - 312q^{98} + O(q^{100}) \)

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
−2.70156
3.70156
−2.70156 3.00000 −0.701562 5.00000 −8.10469 10.1047 23.5078 9.00000 −13.5078
1.2 3.70156 3.00000 5.70156 5.00000 11.1047 −9.10469 −8.50781 9.00000 18.5078
\(n\): 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 1815.4.a.o yes 2
11.b odd 2 1 1815.4.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.4.a.i 2 11.b odd 2 1
1815.4.a.o yes 2 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{2} - T_{2} - 10 \)
\( T_{7}^{2} - T_{7} - 92 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -10 - T + T^{2} \)
$3$ \( ( -3 + T )^{2} \)
$5$ \( ( -5 + T )^{2} \)
$7$ \( -92 - T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 542 - 47 T + T^{2} \)
$17$ \( 2237 + 98 T + T^{2} \)
$19$ \( 576 + 75 T + T^{2} \)
$23$ \( -9038 + 57 T + T^{2} \)
$29$ \( 3530 + 127 T + T^{2} \)
$31$ \( 3852 + 183 T + T^{2} \)
$37$ \( -50860 + 229 T + T^{2} \)
$41$ \( -288 - 18 T + T^{2} \)
$43$ \( -21240 - 186 T + T^{2} \)
$47$ \( 51250 + 615 T + T^{2} \)
$53$ \( 214822 + 927 T + T^{2} \)
$59$ \( -281404 + 380 T + T^{2} \)
$61$ \( 35978 - 509 T + T^{2} \)
$67$ \( 93136 + 1138 T + T^{2} \)
$71$ \( -141524 + 273 T + T^{2} \)
$73$ \( 37904 - 440 T + T^{2} \)
$79$ \( 187882 - 973 T + T^{2} \)
$83$ \( -279000 - 15 T + T^{2} \)
$89$ \( 382592 + 1288 T + T^{2} \)
$97$ \( -26272 + 76 T + T^{2} \)
show more
show less