Properties

Label 1815.4.a.n
Level $1815$
Weight $4$
Character orbit 1815.a
Self dual yes
Analytic conductor $107.088$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 3 q^{3} + ( -4 + \beta ) q^{4} -5 q^{5} + 3 \beta q^{6} + ( 4 - 4 \beta ) q^{7} + ( 4 - 11 \beta ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + \beta q^{2} + 3 q^{3} + ( -4 + \beta ) q^{4} -5 q^{5} + 3 \beta q^{6} + ( 4 - 4 \beta ) q^{7} + ( 4 - 11 \beta ) q^{8} + 9 q^{9} -5 \beta q^{10} + ( -12 + 3 \beta ) q^{12} + ( 44 + 2 \beta ) q^{13} -16 q^{14} -15 q^{15} + ( -12 - 15 \beta ) q^{16} + ( 30 - 44 \beta ) q^{17} + 9 \beta q^{18} + ( 74 + 22 \beta ) q^{19} + ( 20 - 5 \beta ) q^{20} + ( 12 - 12 \beta ) q^{21} + ( -32 - 60 \beta ) q^{23} + ( 12 - 33 \beta ) q^{24} + 25 q^{25} + ( 8 + 46 \beta ) q^{26} + 27 q^{27} + ( -32 + 16 \beta ) q^{28} + ( 96 - 34 \beta ) q^{29} -15 \beta q^{30} + ( 36 - 12 \beta ) q^{31} + ( -92 + 61 \beta ) q^{32} + ( -176 - 14 \beta ) q^{34} + ( -20 + 20 \beta ) q^{35} + ( -36 + 9 \beta ) q^{36} + ( -130 - 112 \beta ) q^{37} + ( 88 + 96 \beta ) q^{38} + ( 132 + 6 \beta ) q^{39} + ( -20 + 55 \beta ) q^{40} + ( -96 + 154 \beta ) q^{41} -48 q^{42} + ( 196 + 124 \beta ) q^{43} -45 q^{45} + ( -240 - 92 \beta ) q^{46} + ( 4 + 216 \beta ) q^{47} + ( -36 - 45 \beta ) q^{48} + ( -263 - 16 \beta ) q^{49} + 25 \beta q^{50} + ( 90 - 132 \beta ) q^{51} + ( -168 + 38 \beta ) q^{52} + ( 334 - 196 \beta ) q^{53} + 27 \beta q^{54} + ( 192 - 16 \beta ) q^{56} + ( 222 + 66 \beta ) q^{57} + ( -136 + 62 \beta ) q^{58} + ( 4 + 240 \beta ) q^{59} + ( 60 - 15 \beta ) q^{60} + ( 146 - 364 \beta ) q^{61} + ( -48 + 24 \beta ) q^{62} + ( 36 - 36 \beta ) q^{63} + ( 340 + 89 \beta ) q^{64} + ( -220 - 10 \beta ) q^{65} + ( -380 + 16 \beta ) q^{67} + ( -296 + 162 \beta ) q^{68} + ( -96 - 180 \beta ) q^{69} + 80 q^{70} + ( 1008 + 44 \beta ) q^{71} + ( 36 - 99 \beta ) q^{72} + ( 272 - 58 \beta ) q^{73} + ( -448 - 242 \beta ) q^{74} + 75 q^{75} + ( -208 + 8 \beta ) q^{76} + ( 24 + 138 \beta ) q^{78} + ( -474 + 306 \beta ) q^{79} + ( 60 + 75 \beta ) q^{80} + 81 q^{81} + ( 616 + 58 \beta ) q^{82} + ( -70 + 426 \beta ) q^{83} + ( -96 + 48 \beta ) q^{84} + ( -150 + 220 \beta ) q^{85} + ( 496 + 320 \beta ) q^{86} + ( 288 - 102 \beta ) q^{87} + ( 186 - 128 \beta ) q^{89} -45 \beta q^{90} + ( 144 - 176 \beta ) q^{91} + ( -112 + 148 \beta ) q^{92} + ( 108 - 36 \beta ) q^{93} + ( 864 + 220 \beta ) q^{94} + ( -370 - 110 \beta ) q^{95} + ( -276 + 183 \beta ) q^{96} + ( -298 + 428 \beta ) q^{97} + ( -64 - 279 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 6q^{3} - 7q^{4} - 10q^{5} + 3q^{6} + 4q^{7} - 3q^{8} + 18q^{9} + O(q^{10}) \) \( 2q + q^{2} + 6q^{3} - 7q^{4} - 10q^{5} + 3q^{6} + 4q^{7} - 3q^{8} + 18q^{9} - 5q^{10} - 21q^{12} + 90q^{13} - 32q^{14} - 30q^{15} - 39q^{16} + 16q^{17} + 9q^{18} + 170q^{19} + 35q^{20} + 12q^{21} - 124q^{23} - 9q^{24} + 50q^{25} + 62q^{26} + 54q^{27} - 48q^{28} + 158q^{29} - 15q^{30} + 60q^{31} - 123q^{32} - 366q^{34} - 20q^{35} - 63q^{36} - 372q^{37} + 272q^{38} + 270q^{39} + 15q^{40} - 38q^{41} - 96q^{42} + 516q^{43} - 90q^{45} - 572q^{46} + 224q^{47} - 117q^{48} - 542q^{49} + 25q^{50} + 48q^{51} - 298q^{52} + 472q^{53} + 27q^{54} + 368q^{56} + 510q^{57} - 210q^{58} + 248q^{59} + 105q^{60} - 72q^{61} - 72q^{62} + 36q^{63} + 769q^{64} - 450q^{65} - 744q^{67} - 430q^{68} - 372q^{69} + 160q^{70} + 2060q^{71} - 27q^{72} + 486q^{73} - 1138q^{74} + 150q^{75} - 408q^{76} + 186q^{78} - 642q^{79} + 195q^{80} + 162q^{81} + 1290q^{82} + 286q^{83} - 144q^{84} - 80q^{85} + 1312q^{86} + 474q^{87} + 244q^{89} - 45q^{90} + 112q^{91} - 76q^{92} + 180q^{93} + 1948q^{94} - 850q^{95} - 369q^{96} - 168q^{97} - 407q^{98} + O(q^{100}) \)

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
−1.56155
2.56155
−1.56155 3.00000 −5.56155 −5.00000 −4.68466 10.2462 21.1771 9.00000 7.80776
1.2 2.56155 3.00000 −1.43845 −5.00000 7.68466 −6.24621 −24.1771 9.00000 −12.8078
\(n\): 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.n 2
11.b odd 2 1 165.4.a.c 2
33.d even 2 1 495.4.a.d 2
55.d odd 2 1 825.4.a.m 2
55.e even 4 2 825.4.c.j 4
165.d even 2 1 2475.4.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.c 2 11.b odd 2 1
495.4.a.d 2 33.d even 2 1
825.4.a.m 2 55.d odd 2 1
825.4.c.j 4 55.e even 4 2
1815.4.a.n 2 1.a even 1 1 trivial
2475.4.a.n 2 165.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1815))\):

\( T_{2}^{2} - T_{2} - 4 \)
\( T_{7}^{2} - 4 T_{7} - 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 - T + T^{2} \)
$3$ \( ( -3 + T )^{2} \)
$5$ \( ( 5 + T )^{2} \)
$7$ \( -64 - 4 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 2008 - 90 T + T^{2} \)
$17$ \( -8164 - 16 T + T^{2} \)
$19$ \( 5168 - 170 T + T^{2} \)
$23$ \( -11456 + 124 T + T^{2} \)
$29$ \( 1328 - 158 T + T^{2} \)
$31$ \( 288 - 60 T + T^{2} \)
$37$ \( -18716 + 372 T + T^{2} \)
$41$ \( -100432 + 38 T + T^{2} \)
$43$ \( 1216 - 516 T + T^{2} \)
$47$ \( -185744 - 224 T + T^{2} \)
$53$ \( -107572 - 472 T + T^{2} \)
$59$ \( -229424 - 248 T + T^{2} \)
$61$ \( -561812 + 72 T + T^{2} \)
$67$ \( 137296 + 744 T + T^{2} \)
$71$ \( 1052672 - 2060 T + T^{2} \)
$73$ \( 44752 - 486 T + T^{2} \)
$79$ \( -294912 + 642 T + T^{2} \)
$83$ \( -750824 - 286 T + T^{2} \)
$89$ \( -54748 - 244 T + T^{2} \)
$97$ \( -771476 + 168 T + T^{2} \)
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