Properties

Label 1805.2.b.e
Level $1805$
Weight $2$
Character orbit 1805.b
Analytic conductor $14.413$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1805.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.4129975648\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.16516096.1
Defining polynomial: \( x^{6} + 9x^{4} + 13x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 9x^{4} + 13x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 8\nu^{2} + 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + \nu^{4} + 10\nu^{3} + 6\nu^{2} + 19\nu - 1 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - \nu^{4} + 10\nu^{3} - 6\nu^{2} + 19\nu + 1 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} - 8\nu^{3} - 7\nu ) / 2 \) 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
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{4} + 8\beta_{3} - 6\beta_{2} + 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -10\beta_{5} - 8\beta_{4} - 8\beta_{3} + 41\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1805\mathbb{Z}\right)^\times\).

\(n\) \(362\) \(1446\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1084.1
1.30397i
2.68667i
0.285442i
0.285442i
2.68667i
1.30397i
2.41987i 0.537080i −3.85577 −2.07772 0.826491i −1.29966 3.18676i 4.49073i 2.71155 −2.00000 + 5.02781i
1084.2 1.82254i 2.31446i −1.32164 1.94827 1.09737i 4.21819 1.45033i 1.23634i −2.35673 −2.00000 3.55080i
1084.3 0.906968i 3.21789i 1.17741 −0.370556 2.20515i −2.91852 2.59637i 2.88181i −7.35482 −2.00000 + 0.336083i
1084.4 0.906968i 3.21789i 1.17741 −0.370556 + 2.20515i −2.91852 2.59637i 2.88181i −7.35482 −2.00000 0.336083i
1084.5 1.82254i 2.31446i −1.32164 1.94827 + 1.09737i 4.21819 1.45033i 1.23634i −2.35673 −2.00000 + 3.55080i
1084.6 2.41987i 0.537080i −3.85577 −2.07772 + 0.826491i −1.29966 3.18676i 4.49073i 2.71155 −2.00000 5.02781i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1084.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1805.2.b.e 6
5.b even 2 1 inner 1805.2.b.e 6
5.c odd 4 2 9025.2.a.bx 6
19.b odd 2 1 95.2.b.b 6
57.d even 2 1 855.2.c.d 6
76.d even 2 1 1520.2.d.h 6
95.d odd 2 1 95.2.b.b 6
95.g even 4 2 475.2.a.j 6
285.b even 2 1 855.2.c.d 6
285.j odd 4 2 4275.2.a.br 6
380.d even 2 1 1520.2.d.h 6
380.j odd 4 2 7600.2.a.ck 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.b.b 6 19.b odd 2 1
95.2.b.b 6 95.d odd 2 1
475.2.a.j 6 95.g even 4 2
855.2.c.d 6 57.d even 2 1
855.2.c.d 6 285.b even 2 1
1520.2.d.h 6 76.d even 2 1
1520.2.d.h 6 380.d even 2 1
1805.2.b.e 6 1.a even 1 1 trivial
1805.2.b.e 6 5.b even 2 1 inner
4275.2.a.br 6 285.j odd 4 2
7600.2.a.ck 6 380.j odd 4 2
9025.2.a.bx 6 5.c odd 4 2

Hecke kernels

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

\( T_{2}^{6} + 10T_{2}^{4} + 27T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{29} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 10 T^{4} + 27 T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{6} + 16 T^{4} + 60 T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{6} + T^{5} - T^{4} - 2 T^{3} - 5 T^{2} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 19 T^{4} + 104 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( (T^{3} - T^{2} - 16 T + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 28 T^{4} + 236 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$17$ \( T^{6} + 59 T^{4} + 1008 T^{2} + \cdots + 5184 \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 36 T^{4} + 208 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( (T - 6)^{6} \) Copy content Toggle raw display
$31$ \( (T^{3} - 56 T - 128)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 56 T^{4} + 764 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$41$ \( (T^{3} + 6 T^{2} - 44 T + 24)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 19 T^{4} + 104 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$47$ \( T^{6} + 187 T^{4} + 7464 T^{2} + \cdots + 85264 \) Copy content Toggle raw display
$53$ \( T^{6} + 156 T^{4} + 2476 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$59$ \( (T^{3} - 10 T^{2} + 8 T + 48)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 7 T^{2} - 104 T - 776)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 340 T^{4} + 28556 T^{2} + \cdots + 484416 \) Copy content Toggle raw display
$71$ \( (T^{3} + 26 T^{2} + 200 T + 432)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 131 T^{4} + 1616 T^{2} + \cdots + 5184 \) Copy content Toggle raw display
$79$ \( (T^{3} + 12 T^{2} - 8 T - 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 228 T^{4} + 11728 T^{2} + \cdots + 141376 \) Copy content Toggle raw display
$89$ \( (T^{3} - 12 T^{2} - 284 T + 3456)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 28 T^{4} + 236 T^{2} + \cdots + 576 \) Copy content Toggle raw display
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