Properties

Label 1815.4.a.r
Level $1815$
Weight $4$
Character orbit 1815.a
Self dual yes
Analytic conductor $107.088$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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: \(3\)
Coefficient field: 3.3.47528.1
Defining polynomial: \(x^{3} - x^{2} - 26 x - 22\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} + 3 q^{3} + ( 10 + \beta_{2} ) q^{4} -5 q^{5} + ( 3 - 3 \beta_{1} ) q^{6} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{7} + ( -3 - 9 \beta_{1} + 2 \beta_{2} ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} + 3 q^{3} + ( 10 + \beta_{2} ) q^{4} -5 q^{5} + ( 3 - 3 \beta_{1} ) q^{6} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{7} + ( -3 - 9 \beta_{1} + 2 \beta_{2} ) q^{8} + 9 q^{9} + ( -5 + 5 \beta_{1} ) q^{10} + ( 30 + 3 \beta_{2} ) q^{12} + ( -38 + 2 \beta_{2} ) q^{13} + ( -28 + 16 \beta_{1} - 6 \beta_{2} ) q^{14} -15 q^{15} + ( 60 - 2 \beta_{1} + 5 \beta_{2} ) q^{16} + ( 34 + 2 \beta_{1} + 2 \beta_{2} ) q^{17} + ( 9 - 9 \beta_{1} ) q^{18} + ( 24 - 14 \beta_{1} + 8 \beta_{2} ) q^{19} + ( -50 - 5 \beta_{2} ) q^{20} + ( -12 + 6 \beta_{1} - 6 \beta_{2} ) q^{21} + ( 48 - 24 \beta_{1} + 4 \beta_{2} ) q^{23} + ( -9 - 27 \beta_{1} + 6 \beta_{2} ) q^{24} + 25 q^{25} + ( -48 + 24 \beta_{1} + 4 \beta_{2} ) q^{26} + 27 q^{27} + ( -238 + 38 \beta_{1} - 12 \beta_{2} ) q^{28} + ( 62 + 34 \beta_{1} ) q^{29} + ( -15 + 15 \beta_{1} ) q^{30} + ( 96 - 40 \beta_{1} - 8 \beta_{2} ) q^{31} + ( 93 - 21 \beta_{1} - 4 \beta_{2} ) q^{32} + ( -10 - 50 \beta_{1} + 2 \beta_{2} ) q^{34} + ( 20 - 10 \beta_{1} + 10 \beta_{2} ) q^{35} + ( 90 + 9 \beta_{2} ) q^{36} + ( 286 - 20 \beta_{1} ) q^{37} + ( 222 - 66 \beta_{1} + 30 \beta_{2} ) q^{38} + ( -114 + 6 \beta_{2} ) q^{39} + ( 15 + 45 \beta_{1} - 10 \beta_{2} ) q^{40} + ( -30 - 66 \beta_{1} ) q^{41} + ( -84 + 48 \beta_{1} - 18 \beta_{2} ) q^{42} + ( -48 + 22 \beta_{1} + 14 \beta_{2} ) q^{43} -45 q^{45} + ( 436 - 52 \beta_{1} + 32 \beta_{2} ) q^{46} + ( 168 - 4 \beta_{2} ) q^{47} + ( 180 - 6 \beta_{1} + 15 \beta_{2} ) q^{48} + ( 117 - 72 \beta_{1} ) q^{49} + ( 25 - 25 \beta_{1} ) q^{50} + ( 102 + 6 \beta_{1} + 6 \beta_{2} ) q^{51} + ( -172 - 4 \beta_{1} - 32 \beta_{2} ) q^{52} + ( 126 - 96 \beta_{1} + 16 \beta_{2} ) q^{53} + ( 27 - 27 \beta_{1} ) q^{54} + ( -600 + 156 \beta_{1} - 14 \beta_{2} ) q^{56} + ( 72 - 42 \beta_{1} + 24 \beta_{2} ) q^{57} + ( -516 - 96 \beta_{1} - 34 \beta_{2} ) q^{58} + ( 172 + 32 \beta_{1} + 8 \beta_{2} ) q^{59} + ( -150 - 15 \beta_{2} ) q^{60} + ( -134 - 12 \beta_{1} - 68 \beta_{2} ) q^{61} + ( 816 + 24 \beta_{2} ) q^{62} + ( -36 + 18 \beta_{1} - 18 \beta_{2} ) q^{63} + ( -10 - 28 \beta_{1} - 27 \beta_{2} ) q^{64} + ( 190 - 10 \beta_{2} ) q^{65} + ( -176 + 100 \beta_{1} - 16 \beta_{2} ) q^{67} + ( 558 + 30 \beta_{1} + 38 \beta_{2} ) q^{68} + ( 144 - 72 \beta_{1} + 12 \beta_{2} ) q^{69} + ( 140 - 80 \beta_{1} + 30 \beta_{2} ) q^{70} + ( -324 + 60 \beta_{1} - 28 \beta_{2} ) q^{71} + ( -27 - 81 \beta_{1} + 18 \beta_{2} ) q^{72} + ( -194 - 36 \beta_{1} + 38 \beta_{2} ) q^{73} + ( 626 - 266 \beta_{1} + 20 \beta_{2} ) q^{74} + 75 q^{75} + ( 1002 - 254 \beta_{1} + 62 \beta_{2} ) q^{76} + ( -144 + 72 \beta_{1} + 12 \beta_{2} ) q^{78} + ( 212 - 94 \beta_{1} - 8 \beta_{2} ) q^{79} + ( -300 + 10 \beta_{1} - 25 \beta_{2} ) q^{80} + 81 q^{81} + ( 1092 + 96 \beta_{1} + 66 \beta_{2} ) q^{82} + ( -60 + 180 \beta_{1} - 54 \beta_{2} ) q^{83} + ( -714 + 114 \beta_{1} - 36 \beta_{2} ) q^{84} + ( -170 - 10 \beta_{1} - 10 \beta_{2} ) q^{85} + ( -492 - 72 \beta_{1} + 6 \beta_{2} ) q^{86} + ( 186 + 102 \beta_{1} ) q^{87} + ( 254 + 28 \beta_{1} + 120 \beta_{2} ) q^{89} + ( -45 + 45 \beta_{1} ) q^{90} + ( -244 - 40 \beta_{1} + 92 \beta_{2} ) q^{91} + ( 776 - 416 \beta_{1} + 84 \beta_{2} ) q^{92} + ( 288 - 120 \beta_{1} - 24 \beta_{2} ) q^{93} + ( 188 - 140 \beta_{1} - 8 \beta_{2} ) q^{94} + ( -120 + 70 \beta_{1} - 40 \beta_{2} ) q^{95} + ( 279 - 63 \beta_{1} - 12 \beta_{2} ) q^{96} + ( 658 + 100 \beta_{1} - 44 \beta_{2} ) q^{97} + ( 1341 - 45 \beta_{1} + 72 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 9 q^{3} + 30 q^{4} - 15 q^{5} + 6 q^{6} - 10 q^{7} - 18 q^{8} + 27 q^{9} + O(q^{10}) \) \( 3 q + 2 q^{2} + 9 q^{3} + 30 q^{4} - 15 q^{5} + 6 q^{6} - 10 q^{7} - 18 q^{8} + 27 q^{9} - 10 q^{10} + 90 q^{12} - 114 q^{13} - 68 q^{14} - 45 q^{15} + 178 q^{16} + 104 q^{17} + 18 q^{18} + 58 q^{19} - 150 q^{20} - 30 q^{21} + 120 q^{23} - 54 q^{24} + 75 q^{25} - 120 q^{26} + 81 q^{27} - 676 q^{28} + 220 q^{29} - 30 q^{30} + 248 q^{31} + 258 q^{32} - 80 q^{34} + 50 q^{35} + 270 q^{36} + 838 q^{37} + 600 q^{38} - 342 q^{39} + 90 q^{40} - 156 q^{41} - 204 q^{42} - 122 q^{43} - 135 q^{45} + 1256 q^{46} + 504 q^{47} + 534 q^{48} + 279 q^{49} + 50 q^{50} + 312 q^{51} - 520 q^{52} + 282 q^{53} + 54 q^{54} - 1644 q^{56} + 174 q^{57} - 1644 q^{58} + 548 q^{59} - 450 q^{60} - 414 q^{61} + 2448 q^{62} - 90 q^{63} - 58 q^{64} + 570 q^{65} - 428 q^{67} + 1704 q^{68} + 360 q^{69} + 340 q^{70} - 912 q^{71} - 162 q^{72} - 618 q^{73} + 1612 q^{74} + 225 q^{75} + 2752 q^{76} - 360 q^{78} + 542 q^{79} - 890 q^{80} + 243 q^{81} + 3372 q^{82} - 2028 q^{84} - 520 q^{85} - 1548 q^{86} + 660 q^{87} + 790 q^{89} - 90 q^{90} - 772 q^{91} + 1912 q^{92} + 744 q^{93} + 424 q^{94} - 290 q^{95} + 774 q^{96} + 2074 q^{97} + 3978 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 26 x - 22\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 17 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 17\)

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
5.97123
−0.906392
−4.06484
−4.97123 3.00000 16.7131 −5.00000 −14.9137 −5.48376 −43.3148 9.00000 24.8561
1.2 1.90639 3.00000 −4.36567 −5.00000 5.71918 22.9186 −23.5738 9.00000 −9.53196
1.3 5.06484 3.00000 17.6526 −5.00000 15.1945 −27.4348 48.8887 9.00000 −25.3242
\(n\): 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.r 3
11.b odd 2 1 165.4.a.e 3
33.d even 2 1 495.4.a.k 3
55.d odd 2 1 825.4.a.r 3
55.e even 4 2 825.4.c.k 6
165.d even 2 1 2475.4.a.t 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.e 3 11.b odd 2 1
495.4.a.k 3 33.d even 2 1
825.4.a.r 3 55.d odd 2 1
825.4.c.k 6 55.e even 4 2
1815.4.a.r 3 1.a even 1 1 trivial
2475.4.a.t 3 165.d even 2 1

Hecke kernels

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

\( T_{2}^{3} - 2 T_{2}^{2} - 25 T_{2} + 48 \)
\( T_{7}^{3} + 10 T_{7}^{2} - 604 T_{7} - 3448 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 48 - 25 T - 2 T^{2} + T^{3} \)
$3$ \( ( -3 + T )^{3} \)
$5$ \( ( 5 + T )^{3} \)
$7$ \( -3448 - 604 T + 10 T^{2} + T^{3} \)
$11$ \( T^{3} \)
$13$ \( 37216 + 3712 T + 114 T^{2} + T^{3} \)
$17$ \( -8448 + 2792 T - 104 T^{2} + T^{3} \)
$19$ \( -65520 - 11496 T - 58 T^{2} + T^{3} \)
$23$ \( 148224 - 10736 T - 120 T^{2} + T^{3} \)
$29$ \( 629760 - 14308 T - 220 T^{2} + T^{3} \)
$31$ \( 9589248 - 38592 T - 248 T^{2} + T^{3} \)
$37$ \( -18607336 + 223548 T - 838 T^{2} + T^{3} \)
$41$ \( 3013632 - 106596 T + 156 T^{2} + T^{3} \)
$43$ \( -1445400 - 44940 T + 122 T^{2} + T^{3} \)
$47$ \( -4372224 + 82192 T - 504 T^{2} + T^{3} \)
$53$ \( -3654264 - 222068 T - 282 T^{2} + T^{3} \)
$59$ \( -1206720 + 57584 T - 548 T^{2} + T^{3} \)
$61$ \( -342344792 - 681332 T + 414 T^{2} + T^{3} \)
$67$ \( -8135552 - 206752 T + 428 T^{2} + T^{3} \)
$71$ \( -2867712 + 97888 T + 912 T^{2} + T^{3} \)
$73$ \( -26458592 - 100544 T + 618 T^{2} + T^{3} \)
$79$ \( 88503440 - 161224 T - 542 T^{2} + T^{3} \)
$83$ \( 434328048 - 1091340 T + T^{3} \)
$89$ \( 1941629400 - 2118532 T - 790 T^{2} + T^{3} \)
$97$ \( 98075336 + 967212 T - 2074 T^{2} + T^{3} \)
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