Properties

Label 1805.2.a.p
Level $1805$
Weight $2$
Character orbit 1805.a
Self dual yes
Analytic conductor $14.413$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1805.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4129975648\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.11344.1
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + \beta_{3} q^{3} + ( - \beta_{2} + \beta_1 + 1) q^{4} - q^{5} + (2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{6} + ( - 2 \beta_1 + 2) q^{7} + ( - \beta_{3} - \beta_{2} + 2) q^{8} + ( - 2 \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + \beta_{3} q^{3} + ( - \beta_{2} + \beta_1 + 1) q^{4} - q^{5} + (2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{6} + ( - 2 \beta_1 + 2) q^{7} + ( - \beta_{3} - \beta_{2} + 2) q^{8} + ( - 2 \beta_{2} + 1) q^{9} + \beta_{2} q^{10} + 2 \beta_1 q^{11} + (3 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{12} + ( - \beta_{3} - 2 \beta_1) q^{13} + (2 \beta_{3} + 2 \beta_1 + 2) q^{14} - \beta_{3} q^{15} + ( - 2 \beta_{3} - 2 \beta_{2} - 1) q^{16} + (2 \beta_{3} + 2) q^{17} + ( - 3 \beta_{2} + 2 \beta_1 + 6) q^{18} + (\beta_{2} - \beta_1 - 1) q^{20} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{21} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{22} + ( - 2 \beta_{3} - 2 \beta_1 - 2) q^{23} + (4 \beta_{3} + 3 \beta_{2} - \beta_1 - 2) q^{24} + q^{25} + (\beta_{2} + 3 \beta_1) q^{26} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 4) q^{27} + (2 \beta_{3} - 2 \beta_{2} - 2) q^{28} + ( - 2 \beta_{3} - 2) q^{29} + ( - 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{30} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{31} + ( - 2 \beta_{3} - \beta_{2} + 4 \beta_1 - 2) q^{32} + (2 \beta_{2} - 2 \beta_1) q^{33} + (4 \beta_{3} - 2 \beta_1 + 4) q^{34} + (2 \beta_1 - 2) q^{35} + ( - 2 \beta_{3} - 7 \beta_{2} + \beta_1 + 5) q^{36} + ( - \beta_{3} - 2 \beta_{2}) q^{37} + (2 \beta_1 - 4) q^{39} + (\beta_{3} + \beta_{2} - 2) q^{40} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{41} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 8) q^{42} + ( - 2 \beta_1 + 2) q^{43} + ( - 2 \beta_{3} + 2 \beta_1 + 4) q^{44} + (2 \beta_{2} - 1) q^{45} + ( - 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{46} + ( - 4 \beta_{2} + 2 \beta_1 - 6) q^{47} + (3 \beta_{3} + 6 \beta_{2} - 2 \beta_1 - 4) q^{48} + (4 \beta_{2} - 4 \beta_1 + 9) q^{49} - \beta_{2} q^{50} + (2 \beta_{3} - 4 \beta_{2} + 8) q^{51} + ( - \beta_{3} - 2 \beta_{2} - 6) q^{52} + (\beta_{3} + 2 \beta_{2} + 4) q^{53} + (6 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{54} - 2 \beta_1 q^{55} + (2 \beta_{2} - 4 \beta_1 + 6) q^{56} + ( - 4 \beta_{3} + 2 \beta_1 - 4) q^{58} + ( - 2 \beta_{2} - 2 \beta_1) q^{59} + ( - 3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{60} + ( - 2 \beta_{2} + 4 \beta_1 + 2) q^{61} + ( - 6 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{62} + (4 \beta_{3} + 2 \beta_1 + 6) q^{63} + ( - 4 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{64} + (\beta_{3} + 2 \beta_1) q^{65} + (2 \beta_{3} + 4 \beta_{2} - 4) q^{66} + (3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 4) q^{67} + (6 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 6) q^{68} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 8) q^{69} + ( - 2 \beta_{3} - 2 \beta_1 - 2) q^{70} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 4) q^{71} + ( - 5 \beta_{3} - 9 \beta_{2} + 4 \beta_1 + 4) q^{72} + ( - 2 \beta_{3} + 6) q^{73} + ( - 2 \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 4) q^{74} + \beta_{3} q^{75} + ( - 4 \beta_{2} - 12) q^{77} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{78} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{79} + (2 \beta_{3} + 2 \beta_{2} + 1) q^{80} + ( - 2 \beta_{2} + 4 \beta_1 + 1) q^{81} + ( - 2 \beta_{3} + 2 \beta_{2} + 4) q^{82} + (2 \beta_{3} - 2 \beta_1 + 2) q^{83} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 12) q^{84} + ( - 2 \beta_{3} - 2) q^{85} + (2 \beta_{3} + 2 \beta_1 + 2) q^{86} + ( - 2 \beta_{3} + 4 \beta_{2} - 8) q^{87} + ( - 2 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 2) q^{88} + (2 \beta_{3} + 4 \beta_{2} + 2) q^{89} + (3 \beta_{2} - 2 \beta_1 - 6) q^{90} + ( - 2 \beta_{3} + 6 \beta_{2} - 2 \beta_1 + 12) q^{91} + ( - 4 \beta_{3} - 2 \beta_{2} - 10) q^{92} + (8 \beta_{2} - 4 \beta_1 - 4) q^{93} + ( - 2 \beta_{3} + 2 \beta_1 + 10) q^{94} + (9 \beta_{2} - 5 \beta_1 - 6) q^{96} + (\beta_{3} + 2 \beta_{2} - 4 \beta_1 - 4) q^{97} + (4 \beta_{3} - \beta_{2} - 8) q^{98} + ( - 4 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{3} + 8 q^{4} - 4 q^{5} + 4 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{3} + 8 q^{4} - 4 q^{5} + 4 q^{7} + 12 q^{8} + 8 q^{9} - 2 q^{10} + 4 q^{11} - 6 q^{12} - 2 q^{13} + 8 q^{14} + 2 q^{15} + 4 q^{16} + 4 q^{17} + 34 q^{18} - 8 q^{20} + 4 q^{21} - 4 q^{22} - 8 q^{23} - 24 q^{24} + 4 q^{25} + 4 q^{26} + 4 q^{27} - 8 q^{28} - 4 q^{29} - 4 q^{31} + 6 q^{32} - 8 q^{33} + 4 q^{34} - 4 q^{35} + 40 q^{36} + 6 q^{37} - 12 q^{39} - 12 q^{40} - 16 q^{41} + 28 q^{42} + 4 q^{43} + 24 q^{44} - 8 q^{45} - 12 q^{47} - 38 q^{48} + 20 q^{49} + 2 q^{50} + 36 q^{51} - 18 q^{52} + 10 q^{53} - 20 q^{54} - 4 q^{55} + 12 q^{56} - 4 q^{58} + 6 q^{60} + 20 q^{61} + 20 q^{62} + 20 q^{63} - 4 q^{64} + 2 q^{65} - 28 q^{66} + 18 q^{67} + 4 q^{68} - 28 q^{69} - 8 q^{70} + 20 q^{71} + 52 q^{72} + 28 q^{73} + 32 q^{74} - 2 q^{75} - 40 q^{77} - 12 q^{78} + 16 q^{79} - 4 q^{80} + 16 q^{81} + 16 q^{82} + 44 q^{84} - 4 q^{85} + 8 q^{86} - 36 q^{87} + 12 q^{88} - 4 q^{89} - 34 q^{90} + 36 q^{91} - 28 q^{92} - 40 q^{93} + 48 q^{94} - 52 q^{96} - 30 q^{97} - 38 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 3\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 5\beta _1 + 4 \) 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
2.78165
−1.51658
−0.552409
1.28734
−1.95594 −0.296842 1.82571 −1.00000 0.580605 −3.56331 0.340899 −2.91188 1.95594
1.2 −0.816594 −1.53844 −1.33317 −1.00000 1.25628 5.03316 2.72185 −0.633188 0.816594
1.3 2.14243 2.87834 2.59002 −1.00000 6.16666 3.10482 1.26409 5.28487 −2.14243
1.4 2.63010 −3.04306 4.91744 −1.00000 −8.00355 −0.574672 7.67316 6.26020 −2.63010
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(19\) \(-1\)

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 1805.2.a.p 4
5.b even 2 1 9025.2.a.bf 4
19.b odd 2 1 95.2.a.b 4
57.d even 2 1 855.2.a.m 4
76.d even 2 1 1520.2.a.t 4
95.d odd 2 1 475.2.a.i 4
95.g even 4 2 475.2.b.e 8
133.c even 2 1 4655.2.a.y 4
152.b even 2 1 6080.2.a.ch 4
152.g odd 2 1 6080.2.a.cc 4
285.b even 2 1 4275.2.a.bo 4
380.d even 2 1 7600.2.a.cf 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.a.b 4 19.b odd 2 1
475.2.a.i 4 95.d odd 2 1
475.2.b.e 8 95.g even 4 2
855.2.a.m 4 57.d even 2 1
1520.2.a.t 4 76.d even 2 1
1805.2.a.p 4 1.a even 1 1 trivial
4275.2.a.bo 4 285.b even 2 1
4655.2.a.y 4 133.c even 2 1
6080.2.a.cc 4 152.g odd 2 1
6080.2.a.ch 4 152.b even 2 1
7600.2.a.cf 4 380.d even 2 1
9025.2.a.bf 4 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1805))\):

\( T_{2}^{4} - 2T_{2}^{3} - 6T_{2}^{2} + 8T_{2} + 9 \) Copy content Toggle raw display
\( T_{3}^{4} + 2T_{3}^{3} - 8T_{3}^{2} - 16T_{3} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} - 6 T^{2} + 8 T + 9 \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} - 8 T^{2} - 16 T - 4 \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 4 T^{3} - 16 T^{2} + 48 T + 32 \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} - 16 T^{2} + 32 T + 48 \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{3} - 24 T^{2} - 32 T + 20 \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} - 32 T^{2} + 16 T + 48 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 8 T^{3} - 24 T^{2} - 176 T + 288 \) Copy content Toggle raw display
$29$ \( T^{4} + 4 T^{3} - 32 T^{2} - 16 T + 48 \) Copy content Toggle raw display
$31$ \( T^{4} + 4 T^{3} - 80 T^{2} - 512 T - 640 \) Copy content Toggle raw display
$37$ \( T^{4} - 6 T^{3} - 24 T^{2} + 40 T + 4 \) Copy content Toggle raw display
$41$ \( T^{4} + 16 T^{3} + 56 T^{2} + \cdots - 240 \) Copy content Toggle raw display
$43$ \( T^{4} - 4 T^{3} - 16 T^{2} + 48 T + 32 \) Copy content Toggle raw display
$47$ \( T^{4} + 12 T^{3} - 64 T^{2} + \cdots + 1056 \) Copy content Toggle raw display
$53$ \( T^{4} - 10 T^{3} + 184 T - 348 \) Copy content Toggle raw display
$59$ \( T^{4} - 64 T^{2} + 224 T - 192 \) Copy content Toggle raw display
$61$ \( T^{4} - 20 T^{3} + 56 T^{2} + \cdots - 2656 \) Copy content Toggle raw display
$67$ \( T^{4} - 18 T^{3} + 8 T^{2} + \cdots - 1076 \) Copy content Toggle raw display
$71$ \( T^{4} - 20 T^{3} + 32 T^{2} + \cdots - 4224 \) Copy content Toggle raw display
$73$ \( T^{4} - 28 T^{3} + 256 T^{2} + \cdots + 176 \) Copy content Toggle raw display
$79$ \( T^{4} - 16 T^{3} + 32 T^{2} + \cdots - 1856 \) Copy content Toggle raw display
$83$ \( T^{4} - 72 T^{2} - 112 T + 480 \) Copy content Toggle raw display
$89$ \( T^{4} + 4 T^{3} - 144 T^{2} + \cdots + 240 \) Copy content Toggle raw display
$97$ \( T^{4} + 30 T^{3} + 224 T^{2} + \cdots - 1388 \) Copy content Toggle raw display
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