Properties

Label 1805.2.a.r
Level $1805$
Weight $2$
Character orbit 1805.a
Self dual yes
Analytic conductor $14.413$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.5822000.1
Defining polynomial: \( x^{6} - x^{5} - 8x^{4} + 5x^{3} + 14x^{2} + x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_1 + 1) q^{2} + ( - \beta_{4} + \beta_{3} + 1) q^{3} + (\beta_{4} - \beta_{2} + 1) q^{4} - q^{5} + ( - \beta_{5} - \beta_{2} + \beta_1 + 1) q^{6} + (\beta_{5} - \beta_1 + 1) q^{7} + (2 \beta_{5} + 3 \beta_{3} - \beta_1 + 1) q^{8} + ( - \beta_{5} - 2 \beta_{4} - \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_1 + 1) q^{2} + ( - \beta_{4} + \beta_{3} + 1) q^{3} + (\beta_{4} - \beta_{2} + 1) q^{4} - q^{5} + ( - \beta_{5} - \beta_{2} + \beta_1 + 1) q^{6} + (\beta_{5} - \beta_1 + 1) q^{7} + (2 \beta_{5} + 3 \beta_{3} - \beta_1 + 1) q^{8} + ( - \beta_{5} - 2 \beta_{4} - \beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_{3} + \beta_1 - 1) q^{10} + ( - \beta_{5} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{11} + (3 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{12} + ( - \beta_{4} - \beta_{2} + 2 \beta_1) q^{13} + (\beta_{5} + 2 \beta_{4} - \beta_{2} - \beta_1 + 3) q^{14} + (\beta_{4} - \beta_{3} - 1) q^{15} + (2 \beta_{5} + \beta_{4} - 3 \beta_{2} + 2 \beta_1 + 1) q^{16} + ( - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{17} + ( - 2 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + \beta_{2} + 1) q^{18} + ( - \beta_{4} + \beta_{2} - 1) q^{20} + (2 \beta_{5} - \beta_{4} - 3 \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{21} + (4 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 6) q^{22} + ( - 2 \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_1 + 2) q^{23} + (\beta_{5} + 2 \beta_{4} - 6 \beta_{3} - \beta_{2} + 3 \beta_1 - 1) q^{24} + q^{25} + ( - 2 \beta_{4} + 4 \beta_{3} - 2) q^{26} + ( - 2 \beta_{5} - 3 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 5) q^{27} + (2 \beta_{5} + 2 \beta_{4} + 7 \beta_{3} - \beta_{2} - 4 \beta_1 + 5) q^{28} + ( - \beta_{5} - 2 \beta_{3} + \beta_{2} - 3 \beta_1 - 3) q^{29} + (\beta_{5} + \beta_{2} - \beta_1 - 1) q^{30} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{31} + (2 \beta_{5} + 7 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 1) q^{32} + ( - \beta_{5} - 4 \beta_{4} + 5 \beta_{3} - \beta_{2} - \beta_1 + 2) q^{33} + ( - 2 \beta_{5} - 4 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{34} + ( - \beta_{5} + \beta_1 - 1) q^{35} + ( - 3 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{36} + (\beta_{4} + \beta_{2} + 4) q^{37} + ( - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 4) q^{39} + ( - 2 \beta_{5} - 3 \beta_{3} + \beta_1 - 1) q^{40} + ( - \beta_{5} + \beta_{2} + \beta_1 + 2) q^{41} + (2 \beta_{5} + 2 \beta_{3} + \beta_{2} - \beta_1 - 4) q^{42} + (\beta_{5} - \beta_{3} + \beta_{2} + \beta_1 + 4) q^{43} + (4 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 6) q^{44} + (\beta_{5} + 2 \beta_{4} + \beta_{2} - \beta_1 - 1) q^{45} + ( - \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_1 + 4) q^{46} + (2 \beta_{5} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{47} + (4 \beta_{5} - 2 \beta_{4} + \beta_{3} + 3 \beta_{2} - 4 \beta_1 - 3) q^{48} + (2 \beta_{5} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{49} + (\beta_{3} - \beta_1 + 1) q^{50} + ( - \beta_{5} + \beta_{4} + 3 \beta_{3} + \beta_1) q^{51} + ( - 2 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 2) q^{52} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 4) q^{53} + ( - 4 \beta_{5} - \beta_{4} + 4 \beta_{3} + \beta_{2} - 2 \beta_1 + 9) q^{54} + (\beta_{5} + \beta_{3} + \beta_{2} + \beta_1 - 2) q^{55} + (3 \beta_{5} + 2 \beta_{4} + 6 \beta_{3} - 7 \beta_{2} + \beta_1 + 9) q^{56} + ( - 2 \beta_{5} + 2 \beta_{4} - 9 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 1) q^{58} + (\beta_{5} - 3 \beta_{3} + \beta_{2} - \beta_1 - 8) q^{59} + ( - 3 \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{60} + (3 \beta_{5} + \beta_{2} - 3 \beta_1 + 1) q^{61} + ( - 2 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{62} + (4 \beta_{5} - 4 \beta_{4} - 6 \beta_{3} + \beta_{2} + 3 \beta_1 - 8) q^{63} + (3 \beta_{4} + 4 \beta_{3} - 3 \beta_{2} + 9) q^{64} + (\beta_{4} + \beta_{2} - 2 \beta_1) q^{65} + ( - 4 \beta_{5} - 4 \beta_{2} + 6 \beta_1 + 6) q^{66} + (\beta_{5} + \beta_{4} + 3 \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{67} + ( - 2 \beta_{5} - 2 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} - 4) q^{68} + ( - 6 \beta_{5} + 2 \beta_{4} + 8 \beta_{3} - 2 \beta_1 + 5) q^{69} + ( - \beta_{5} - 2 \beta_{4} + \beta_{2} + \beta_1 - 3) q^{70} + ( - \beta_{4} - 2 \beta_{3} + 5 \beta_{2} - 2 \beta_1 + 2) q^{71} + (2 \beta_{5} - 2 \beta_{4} - 7 \beta_{3} + 5 \beta_{2} - 2 \beta_1 - 13) q^{72} + (2 \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{73} + (2 \beta_{3} - 4 \beta_1 + 2) q^{74} + ( - \beta_{4} + \beta_{3} + 1) q^{75} + (3 \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 4) q^{77} + ( - 4 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 6) q^{78} + (3 \beta_{4} - 2 \beta_{3} - \beta_{2} + 2) q^{79} + ( - 2 \beta_{5} - \beta_{4} + 3 \beta_{2} - 2 \beta_1 - 1) q^{80} + ( - 2 \beta_{5} - 5 \beta_{4} + 8 \beta_{3} - \beta_{2} - 2 \beta_1 + 12) q^{81} + ( - 2 \beta_{5} - 2 \beta_{4} + \beta_{2} - \beta_1 - 2) q^{82} + ( - 2 \beta_{5} - 3 \beta_{4} + 4 \beta_{3} + 3 \beta_{2} + 2) q^{83} + ( - 3 \beta_{5} + 5 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 3 \beta_1) q^{84} + (\beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{85} + (2 \beta_{3} - 4 \beta_1) q^{86} + (2 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 6) q^{87} + (6 \beta_{5} - 2 \beta_{4} + 10 \beta_{3} - 2 \beta_{2} - 6 \beta_1 - 6) q^{88} + ( - 2 \beta_{4} + 5 \beta_{3} - 2 \beta_{2} + 4) q^{89} + (2 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - \beta_{2} - 1) q^{90} + (3 \beta_{5} - 4 \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 - 6) q^{91} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{92} + (2 \beta_{5} - 5 \beta_{4} + 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{93} + (\beta_{5} + \beta_{4} - 6 \beta_{3} + \beta_1 - 8) q^{94} + ( - 3 \beta_{5} + 4 \beta_{4} - 6 \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 1) q^{96} + (2 \beta_{5} - \beta_{4} - 5 \beta_{2} - 4) q^{97} + (4 \beta_{5} + 4 \beta_{4} + 5 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 9) q^{98} + ( - 4 \beta_{5} - 6 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 6 \beta_1 + 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} + 4 q^{3} + 8 q^{4} - 6 q^{5} + 8 q^{6} + 7 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{2} + 4 q^{3} + 8 q^{4} - 6 q^{5} + 8 q^{6} + 7 q^{7} + 10 q^{9} - 2 q^{10} + 15 q^{11} - 20 q^{12} + 6 q^{13} + 20 q^{14} - 4 q^{15} + 20 q^{16} + 5 q^{17} - 8 q^{18} - 8 q^{20} + 7 q^{21} + 14 q^{22} + 18 q^{24} + 6 q^{25} - 22 q^{26} + 28 q^{27} + 10 q^{28} - 20 q^{29} - 8 q^{30} + 12 q^{31} - 8 q^{32} + q^{33} - 7 q^{35} - 14 q^{36} + 20 q^{37} + 16 q^{39} + 8 q^{41} - 30 q^{42} + 27 q^{43} + 46 q^{44} - 10 q^{45} + 16 q^{46} - 8 q^{47} - 24 q^{48} + 11 q^{49} + 2 q^{50} - 11 q^{51} + 4 q^{52} + 19 q^{53} + 30 q^{54} - 15 q^{55} + 62 q^{56} + 20 q^{58} - 41 q^{59} + 20 q^{60} + 6 q^{61} - 30 q^{62} - 18 q^{63} + 48 q^{64} - 6 q^{65} + 46 q^{66} - 15 q^{67} - 14 q^{68} - 10 q^{69} - 20 q^{70} + 2 q^{71} - 68 q^{72} + 8 q^{73} + 2 q^{74} + 4 q^{75} + 21 q^{77} + 46 q^{78} + 18 q^{79} - 20 q^{80} + 50 q^{81} - 18 q^{82} - 10 q^{83} + 4 q^{84} - 5 q^{85} - 10 q^{86} - 14 q^{87} - 52 q^{88} + 17 q^{89} + 8 q^{90} - 23 q^{91} + 28 q^{92} + 10 q^{93} - 28 q^{94} + 26 q^{96} - 4 q^{97} + 50 q^{98} + 58 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 8x^{4} + 5x^{3} + 14x^{2} + x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 5\nu^{2} - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - \nu^{4} - 7\nu^{3} + 5\nu^{2} + 9\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 12\nu^{2} + 9\nu - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + \nu^{4} + 9\nu^{3} - 5\nu^{2} - 19\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} + \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{3} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{4} - 5\beta_{3} + 7\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{5} + 9\beta_{3} + 2\beta_{2} + 26\beta _1 + 9 \) 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
2.03780
2.47848
−0.343361
0.228906
−2.31247
−1.08935
−2.65583 −2.41205 5.05344 −1.00000 6.40599 −2.14656 −8.10942 2.81796 2.65583
1.2 −0.860448 0.867394 −1.25963 −1.00000 −0.746348 −0.263921 2.80474 −2.24763 0.860448
1.3 −0.274673 3.09431 −1.92455 −1.00000 −0.849925 3.63772 1.07797 6.57477 0.274673
1.4 1.38913 3.31798 −0.0703236 −1.00000 4.60910 −1.97948 −2.87594 8.00899 −1.38913
1.5 1.69444 −0.918335 0.871115 −1.00000 −1.55606 3.12687 −1.91282 −2.15666 −1.69444
1.6 2.70739 0.0506943 5.32995 −1.00000 0.137249 4.62536 9.01547 −2.99743 −2.70739
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.r yes 6
5.b even 2 1 9025.2.a.bq 6
19.b odd 2 1 1805.2.a.q 6
95.d odd 2 1 9025.2.a.ca 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.2.a.q 6 19.b odd 2 1
1805.2.a.r yes 6 1.a even 1 1 trivial
9025.2.a.bq 6 5.b even 2 1
9025.2.a.ca 6 95.d odd 2 1

Hecke kernels

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

\( T_{2}^{6} - 2T_{2}^{5} - 8T_{2}^{4} + 16T_{2}^{3} + 7T_{2}^{2} - 14T_{2} - 4 \) Copy content Toggle raw display
\( T_{3}^{6} - 4T_{3}^{5} - 6T_{3}^{4} + 28T_{3}^{3} + 4T_{3}^{2} - 20T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 2 T^{5} - 8 T^{4} + 16 T^{3} + \cdots - 4 \) Copy content Toggle raw display
$3$ \( T^{6} - 4 T^{5} - 6 T^{4} + 28 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 7 T^{5} - 2 T^{4} + 75 T^{3} + \cdots - 59 \) Copy content Toggle raw display
$11$ \( T^{6} - 15 T^{5} + 53 T^{4} + \cdots - 4496 \) Copy content Toggle raw display
$13$ \( T^{6} - 6 T^{5} - 19 T^{4} + 180 T^{3} + \cdots + 304 \) Copy content Toggle raw display
$17$ \( T^{6} - 5 T^{5} - 39 T^{4} + 116 T^{3} + \cdots - 16 \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 68 T^{4} + 152 T^{3} + \cdots + 61 \) Copy content Toggle raw display
$29$ \( T^{6} + 20 T^{5} + 71 T^{4} + \cdots - 17236 \) Copy content Toggle raw display
$31$ \( T^{6} - 12 T^{5} + 11 T^{4} + 176 T^{3} + \cdots - 16 \) Copy content Toggle raw display
$37$ \( T^{6} - 20 T^{5} + 147 T^{4} + \cdots - 304 \) Copy content Toggle raw display
$41$ \( T^{6} - 8 T^{5} - 12 T^{4} + 178 T^{3} + \cdots + 149 \) Copy content Toggle raw display
$43$ \( T^{6} - 27 T^{5} + 273 T^{4} + \cdots + 496 \) Copy content Toggle raw display
$47$ \( T^{6} + 8 T^{5} - 76 T^{4} - 464 T^{3} + \cdots + 29 \) Copy content Toggle raw display
$53$ \( T^{6} - 19 T^{5} + 97 T^{4} + \cdots - 4544 \) Copy content Toggle raw display
$59$ \( T^{6} + 41 T^{5} + 639 T^{4} + \cdots + 6416 \) Copy content Toggle raw display
$61$ \( T^{6} - 6 T^{5} - 181 T^{4} + \cdots + 11584 \) Copy content Toggle raw display
$67$ \( T^{6} + 15 T^{5} - 14 T^{4} + \cdots - 961 \) Copy content Toggle raw display
$71$ \( T^{6} - 2 T^{5} - 283 T^{4} + \cdots - 114224 \) Copy content Toggle raw display
$73$ \( T^{6} - 8 T^{5} - 69 T^{4} + 572 T^{3} + \cdots + 496 \) Copy content Toggle raw display
$79$ \( T^{6} - 18 T^{5} + 9 T^{4} + 884 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$83$ \( T^{6} + 10 T^{5} - 279 T^{4} + \cdots + 24304 \) Copy content Toggle raw display
$89$ \( T^{6} - 17 T^{5} - 2 T^{4} + \cdots - 4139 \) Copy content Toggle raw display
$97$ \( T^{6} + 4 T^{5} - 305 T^{4} + \cdots - 198704 \) Copy content Toggle raw display
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