Properties

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

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
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.

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

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

Hecke kernels

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

\( T_{2}^{2} + T_{2} - 10 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} - 92 \) Copy content Toggle raw display

Hecke characteristic polynomials

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